Normat 61:2, 133–152 (2013) 133

Werner Fenchel, a pioneer in convexity theory

and a migrant scientist

Christer Oscar Kiselman

Uppsala University,

Department of Information Technology

P. O. Box 337, SE-751 05 Uppsala

kiselman@it.uu.se, christer@kiselman.eu

www.cb.uu.se/˜kiselman

1 Introduction

Werner Fenchel (1905–1988) was a pioneer in convexity theory and in particular

the use of duality there. When asked about his views on the many terms used to

express this duality he described in a private le tter (1977) the whole development

from Legendre and onwards, as well as his preferences concerning the choice of

terms. The background for his leaving Germany and moving to Denmark and later

to Swe den is sketched in Section 12.

2 Convex sets and convex functions

In a vector space E over the ﬁeld R of real numbers—think of E as the two-

dimensional plane, identiﬁed with R

, or three-dimensional space, identiﬁed with

—we deﬁne, given two points a and b,thesegment with endpoints a and b:

[a, b]={(1 ≠ t)a + tb; t œ R, 0 6 t 6 1}.

A subset A of E is said to be convex if

{a, b} µ A implies that [ a, b] µ A.

The modern theory of convex sets starts with the work of Hermann Minkowski

(1864–1909); see his book (1910), a large part of which was published in 1896.

134 Christer Oscar Kiselman Normat 2/2013

When calculating with functions, it will be convenient to allow inﬁnite values,

thus to let functions take values in R ﬁ {≠Œ} ﬁ {+Œ}, adding two inﬁnities to R.

This is the set of extended real numbers, and will be denoted by R

or [≠Œ, +Œ].

It is ordered so that ≠Œ <x<+Œ for all x œ R.

The operation of addition,

R ◊ R – (x, y) ‘æ x + y œ R,

will cause diﬃculties when we try to deﬁne sums like (+Œ)+(≠Œ). A convenient

solution, pioneered by Moreau (1963; 1966–1967:9) is to extend it in two diﬀerent

ways to operations R

◊ R

æ R

, viz. as upper addition x +

y, deﬁned as +Œ if

one of the terms is equal to +Œ, and lower addition x +

y, deﬁned as ≠Œ if one

of the terms is equal to ≠Œ. (Some authors allow only values in ]≠Œ, +Œ],but

then the rules for avoiding ≠Œ become complicated: if f is admissible, maybe ≠f

is not; the inﬁmum of an admissible function may not be admissible, etc.)

It is convenient to deﬁne convex functions with the help of convex sets. We

deﬁne the ﬁnite epigraph of a function f : E æ R

epi

(f)={(x, t) œ E ◊ R; t > f(x)} µ E ◊ R.

Then a function f : E æ R

is deﬁned to be convex when epi

(f) is convex.

In the other direction we note that a set A is convex if and only its indicator

function indf

is convex. Here indf

: E æ R

is deﬁned to take the value 0 in A

and +Œ in its complement. We have epi

(indf

)=A ◊ [0, +Œ[.

So far, there is no duality.

The book by Bonnesen and Fenchel (1934) is mainly concerned with convex

bodies; it mentions the supporting function but does not go into the duality theory.

Also—remarkably—Fenchel’s survey article (1983), a translation of a talk given in

1973, does not mention duality.

It is tempting to believe that convex sets possess some kind of regularity. But

this is not so. Let W be any subset of R

, as irregular as you like, for instance a

set which is not measurable in the sense of Lebesgue (if you believe in the axiom

of choice). Then the set

{x œ R

; ÎxÎ

< 1} ﬁ {x œ W ; ÎxÎ

=1},

where we use the Euclidean norm Î · Î

, is convex.

3 Duality in convexity theory

The deﬁnition of a convex set is done from the inside and does not need duality,

but there is also a deﬁnition from the outside, which requires an introduction of

duality.

We introduce the algebraic dual of E, which is the set of all linear forms › : E æ

R and is denoted by E

. An elem ent of E

is thus a function › : E æ R such

that ›(x + ty)=›(x)+t›(y) for all elements x, y of E and all real numbers t.We

Normat 2/2013 Christer Oscar Kiselman 135

can now speak of hyperplanes in E: they are given by an equation ›(x)=c for a

nonzero element › of E

and a constant c œ R, as well as aﬃne functions on E;

they are of the form f (x)=›(x)+c.

If E has a topology, we also speak about the dual of E, meaning the set of

all continuous linear forms on E and denoted by E

.IfE is equal to R

with

the usual topology, we have E

= E

. Every linear form can then be written as

x ‘æ › · x = ›

+ ···+ ›

for some vector (›

,...,›

) œ R

By duality in convexity theory I mean any consideration involving the dual or

the algebraic dual to a given space. A typical example is that of a norm Î · Î

dual

to a give n norm Î · Î and deﬁned by

Î›Î

=sup

xœE

ÎxÎ61

›(x), › œ E

If A is a closed convex subset of R

, then it can be described as the intersection

of a family of close d half spaces. Here a closed half space is of the form

{x œ R

; › · x 6 t}

for some › œ R

r {0} and som e real numbe r t. This result is essentially the

Hahn–Banach theorem in the ﬁnite-dimensional case. It follows that if A is an

open convex subset of R

it can be described as the intersection of a family of

open half spaces. An open half space is of the form {x œ R

; › · x<t}.

In general we need half spaces of a more general kind. Let us say that Y is a

reﬁned half space if it is convex and satisﬁes

{x œ R

; › · x<t} µ Y µ {x œ R

; › · x 6 t}.

This means that Y is the union of the open half space and a convex subset A

›

the hyperplane given by the equation › · x = t. Then any convex subset of R

can

be described as the intersection of a family of reﬁned half spaces Y

›

, › of norm 1,

taking A

›

as the set of points y in A which satisfy › · y = t. (Cf. the notion of

reﬁned digital hyperplane in my paper 2004:456, Deﬁnition 6.2.)

4 The Legendre transformation

The Legendre transformation is named for Adrien-Marie Legendre (1752–1833),

who introduced it in classial mechanics to go from the Lagrangian

to the Hamil-

tonian

formulation: the Hamiltonian is, in modern language, the Legendre trans-

form of the Lagrangian (see, e.g., Goldstein 1950:217).

If f : R

æ R is a diﬀerentiable mapping with gradient grad f : R

æ R

,we

may take › = (grad f )(x) as a new independent variable, thus looking at the inverse

(grad f)(x)=› ‘æ x = (grad f)

≠1

(›).

Refers to Joseph-Louis Lagrange (1736–1813).

Refers to William Rowan Hamilton (1805–1865).

136 Christer Oscar Kiselman Normat 2/2013

This is in general a multivalued mapping, which, in Legendre’s time, was no big

deal. The Legendre transform of f is the mapping

– › ‘æ › ·

(grad f)

≠1

(›)

≠ f((grad f)

≠1

(›)) = › · x ≠ f(x).

(See Hiriart-Urruty & Lemaréchal 2001:209.) If f is strictly convex, then x is

uniquely determined by ›, and we arrive at the Fenchel transform (see Section 6

below). If f is convex, the point x is not uniquely determined by ›: the expression

› · x ≠ f(x) can attain its maximum for several choices of x.

5 Tropicalization

Tropicalization means, roughly speaking, to replace a sum or an integral by a

supremum. A simple example is the l

-norm,

ÎxÎ

j=1

1/p

,xœ R

, 1 6 p<+Œ,

which bec ome s

ÎxÎ

sup

j=1,...,n

1/p

=sup

j=1,...,n

when the sum is replaced by the supremum. We shall see that the Fenchel trans-

formation is a case in point.

Tropical geometry is now a ﬁeld of growing interest; see, e.g., Richter-Gebert et

al. (2005) and Maclagan & Sturmfels (2015).

6 The Fenchel transformation

The Fenchel transform

f of a function f : R

æ R

is deﬁned as

f(›)= sup

xœR

› · x ≠ f(x)

, › œ R

Clearly › · x ≠ f(x) 6

f(›), which can be written as

› · x 6 f(x)+

f(›), (›,x) œ R

◊ R

called Fenchel’s inequality. I t follows that the second transform

f satisﬁes

f 6 f.

The operation f ‘æ

f is a cleistomorphism (closure operator) if we deﬁne the

order b etween functions by inclusion of their ﬁnite epigraphs, which is actually a

very natural thing to do. This is because +Œ corresponds to vacuum since its ﬁnite

epigraph is empty, while ≠Œ corresponds to an inﬁnitely dense black hole—in the

model for material objects, the density is e

≠f

Normat 2/2013 Christer Oscar Kiselman 137

On the other hand, if we order functions by the relation f(x) 6 g(x) for all x,

then the operation is an anoiktomorphism. In this way, the Fenchel transformation

enters the ﬁeld of mathematical morphology with its complete lattices, ethmomor-

phisms, cleistomorphisms and anoiktomorphisms.

The e quality

f = f holds if and only if f is convex, lower semicontinuous, and

takes the value ≠Œ only if it is ≠Œ everywhere. Let us denote by CVX

) the

family of all functions with these three prop e rties. The restriction of the Fenchel

transformation to CVX

) is thus an involution, i.e., it is equal to its own inverse.

Artstein-Avidan & Milman (2009:662) have proved that any decreas ing involution

CVX

) æ CVX

) is, up to some obvious modiﬁcations, the restriction to

CVX

) of the Fenchel transformation. Thus the Fenchel transformation is,

essentially, unique among all transformations with the indicated properties. While

this is, intuitively, hardly surprising, it is a very nice property and a convincing

witness to the importance of Fenchel’s work.

If f is deﬁned on an arbitrary vector space E, we deﬁne the Fenchel transform

on its algebraic dual E

f(›)=sup

xœE

›(x) ≠ f(x)

, › œ E

Again,

›(x) 6 f(x)+

f(›), (›,x) œ E

◊ E.

We can apply the transformation a second time, but then it is convenient to choose

ﬁrst an arbitrary subspace  of E

, for instance  = E

, and deﬁne

f(x)=sup

›œ

›(x) ≠

f(›)

,xœ E.

The Fenchel transformation f ‘æ

f is a tropical counterpart of the Laplace

transformation. The Laplace transform of a function g :[0, +Œ[ æ [0, +Œ[ is

(L g)(›)=

⁄

g(x)e

≠›x

dx, › œ R.

If we replace the integral by a supremum and take the logarithm, we get

trop

g)(›)=sup

(log g(x) ≠ ›x)=

f(≠›), › œ R,f= ≠ log g.

7 The supporting function

A special case of the Fenchel transform is obtained when the function is the indi-

cator function of some set: f = indf

.Then

f is the supporting function of A,

denoted by H

f(›)=H

(›)=sup

xœA

› · x, › œ R

138 Christer Oscar Kiselman Normat 2/2013

This function was introduced already by Minkowski (1903:448).

However, conversely, a Fenchel transform

f is also a supporting function if we

go up in dimension. We equip R

◊ R with the inner product

(›, ·) · (x, t)=› · x + ·t, (›, · ), (x, t) œ R

◊ R.

Then

f(›)=H

epi

(f)

(›, ≠1), › œ R

8 Galois connections

Let X and Y be two ordered sets. A Galois connection

is a pair (F, G) of decreasing

mappings F : X æ Y and G : Y æ X such that G ¶ F is larger than the identity in

X and F ¶ G is larger than the identity in Y . Taking both F and G as the Fenchel

transformation, we see that the two form a Galois connection, provided we order

the functions using the inclusion relation between the ﬁnite epigraphs. (See, e.g.,

my paper 2010: Section 4.)

9 Werner Fenchel

Fenchel was born on 1905 May 03 in Berlin. He obtained his doctoral degree

in 1928 at the Friedrich-Wilhelms-Universität in B erlin with Ludwig Bieberbach

(1886–1982) as advisor (Segal 2003:410).

A pape r based on his thesis and enti-

tled “Über Krümmung und Windung geschlossener Raumkurven” appeared in the

Mathematische Annalen in 1929.

After his PhD, Werner Fenchel became an assistant at Göttingen University. He

was dismissed from this position in 1933. He accepted an invitation to Copenhagen

and arrived to Denmark in the summer of 1933. (Jessen 1987:89.)

In Decembe r 1933 he married Käte Sperling (1905–1983), who was also a mathe-

matician, and who had been ousted from a position as high-school teacher in Berlin.

In Denmark, Werner Fenchel was ﬁrst supported by various foundations until he

got a teaching position in 1938 and became a lecturer at Copenhagen University

in 1942. (Jessen 1987:89.)

Named for Évariste Galois (1811–1832).

Bieberbach joined the National Socialist Lecturers Association in November 1933 and a lit-

tle later became the representative of this organization at the University of Berlin (Mehrtens

1987:220). He joined the Sturmabteilung (SA) on 1934 November 05 and the Nationalsozialistische

Deutsche Arbeiterpartei (NSDAP) on 1937 May 01 (Bieberbach’s p ersonal ﬁle in the archive at

Humb oldt University, signature UK B 220, folio 64; information provided by Reinhard Siegmund-

Schultze on 2016 January 26). In a lecture in 1934 he justiﬁed the Nazi-organized boycott of

Edmund Landau’s classes (Mehrtens 1987:227). However, in 1928 he could still serve as advisor

for Fenchel: “no one had ever observed antisemitic tendencies in Bieberbach before the summer

of 1933” (Schappacher 1998:130). For more on Bieberbach, see Segal (2003).

Normat 2/2013 Christer Oscar Kiselman 139

Moritz Werner Fenchel (1905 May 03 — 1988 January 24)

As it turned out, Copenhagen was not suﬃciently far away: on 1940 April 09,

Germany invaded Denmark. However, during the ﬁrst years of the occupation,

Jews were not deported from Denmark; deportations started in October 1943. A

majority of all Jews in Denmark could escape to Sweden, although some were

deported to Theresienstadt (Jacoby 2015). Werner Fenchel came to Sweden in

October 1943 (Jessen 1987:89).

A Danish scho ol, den danske Skole i Lund, started on 1943 November 15 in Lund

in southern Sweden and functioned during 19 months, to the end of the war. Werner

Fenchel and many others taught there, for e xample Harald Bohr. (Siegmund-

Schultze 2009:107, 136.) Also his wife Käte Fenchel taught at the Danish school,

and Werner Fenchel lectured also at Lund University (Jessen 1987:89).

In 1950/51 Werner Fenchel was at the Institute for Advanced Study in Princeton,

NJ, and met Harold W. Kuhn (1925–2014) and Albert W. Tucker (1905–1995)

there:

It must therefore have been very exciting for Kuhn and Tucker when they became

aware that Werner Fenchel from Copenhagen University, who was the leading expert

on convexity at the time, was visiting the Institute for Advanced Study in Princeton

as part of a sabbatical year in the USA in 1950/51. Tucker invited Fenchel to give

a series of lectures on the theory of convexity at the mathematics department at

Princeton University within his ONR pro ject. (Kjeldsen 2010:3250)

Fenchel wrote notes (1951) based on his lectures in Princeton, expanding his short

paper (1949):

Fenchel was motivated by problems and connections in pure mathematics, and his

lecture notes became highly inﬂuential for further developments within the theory

of convexity, especially through the work of R. T. Rockafellar [50].

(Kjeldsen

2010:3251)

Werner Fenchel died on 1988 January 24.

The number [50] refers to Rockafellar (1970). This book the author “dedicated to Fenchel,

as honorary co-author” (Rockafellar 1970:viii).

140 Christer Oscar Kiselman Normat 2/2013

Werner Fenchel lecturing in Menton in the middle of the middle of the sixties

10 Werner Fenchel’s letter

I wrote a letter (in Swedish) to Werner Fenchel on 1977 February 24, asking him

about the history of duality in convexity theory and the many terms used in that

theory. He replied with a letter dated 1977 March 07 (3 pages; in Danish).

reported in detail on the development of convexity theory and the duality which

had been known under so many diﬀerent names, and where he himself had been a

pioneer. I translate it into English below.

Á — Ë

professor w. fenchel 7. 3. 1977

sønderengen 110

2860 søborg

Dear Kiselman.

Thanks for the letter! It is not e asy for me to answer. I have again after many

years looked at the relevant literature, which has delayed my answer. I brieﬂy

report on the history of the matter.

The classical Legendre transformation was introduced purely formally by

Legendre. That it is an involution and closely related to duality was, how-

ever, as far as I know, ﬁrst discovered by Monge somewhat later. One shall of

course keep the name Legendre transformation.

For the beneﬁt of young readers, let me mention that Fenchel’s letter as well as mine were

written on paper using a kind of mechanical device called “typewriter,” which had “arms” or

“bars” equipped with proﬁles in the form of letters, upper case letters being above the corre-

sp onding lower case letter, like this:

.Atinymomentbeforeanarmhitthepaper,atextile

ribb on, slightly moistened with a kind of ink, jumped up in front of the paper and because of

the pressure from the arm left a mark on the paper, very much looking like a letter or symbol,

although often a little fuzzy.

Normat 2/2013 Christer Oscar Kiselman 141

W. H. Young has in a s mall paper from 1912 proved the following. Let Ï(x)

be positive and diﬀerentiable with Ï

(x) > 0 for x>0, and let Â (y) be the

inverse of Ï.Thenwehavewithb = Ï(a),

xy ≠ ab 6

⁄

Ï(›)d› +

⁄

Â(÷)d÷.

If we put f(x)=

Ï(›)d› and

f(y)=ab +

Â(÷)d÷, we easily see that f ‘æ

is a special case of the classical Legendre transformation, which Young, however,

did not notice, even though this is clear from his proof. He is only interested in

the inequality.

The next contribution to the topic is due to Z. W. Birnbaum & W. Orlicz,

1931. They consider convex functions f(x) which are deﬁned for x œ R and

satisfy f (0) = 0, f (x) > 0 for x>0, f (≠x)=f (x), f(x)/x æ 0 as x æ 0,

f(x)/x æŒas x æŒ. Here the “complementary” function

f to f is deﬁned

(ú)

f(y)=sup

xœR

(xy ≠f(x)),

and it is shown that

f = f. The authors also show that f (x)=

Ï(›)d› and

f(y)=

Â(÷)d÷,whereÏ is non-decreasing and continuous to the right, and

Â is its inverse, conveniently deﬁned with the same properties.

In 1939 there appeared a somewhat imprecise note by S. Mandelbrojt,

where

he deﬁned

f by (ú) for an arbitrary convex function of one variable, but without

investigating under which conditions the transformation f ‘æ

f is an involution.

It is clear that he did not know about Birnbaum & Orlicz.

When I wrote my paper “On conjugate convex functions” in 1949, I also did

not know about Birnbaum & Orlicz. I deﬁne there

f for an arbitrary convex

function deﬁned in a convex subset C of R

f(y)=sup

xœC

(Èx, yÍ≠f (x))

and ﬁnd a neces sary and suﬃcient condition for the equality

f = f to hold. The

inequality

(úú) Èx, yÍ 6 f(x)+

f(y)

is a trivial consequence. I also show that f ‘æ

f under suitable diﬀerentiability

hypothes es is the Legendre transformation. In Lecture Notes, Princeton 1951,

I have given a more systematic treatment of the transformation and s everal

variants. A further development is due to R. T. Rockafellar from 1963.

Generalizations to inﬁnite-dimensional spaces have been undertaken inde-

pendently by J. J. Moreau, 1962, A. Brøndsted, 1964, and U. Dieter, 1965.

That his note is imprecise, as Fenchel writes, is due to the fact that Mandelbrojt assumes

real values of the functions without noticing that, if f has real values and does not grow faster

than all linear functions, then

f(y) will not be real valued; it will take the value +Œ when y or

≠y is large.

142 Christer Oscar Kiselman Normat 2/2013

In the bibliography in Rockafellar’s book “Convex Analysis” all the mentioned

papers are listed.

Now to the question of the name! I admit that the situation is rather chaotic.

That I called

f the function conjugated to f depends on the fact that I considered

the relation as a generalization of the “conjugated expononents” of classical

analysis x

/p and y

/q with 1/p +1/q =1. This was also the starting point

for Young and Birnbaum–Orlicz. But the connection with that has become

very weak and the name is therefore not adequate. Rockafellar and Brøndsted

have retained my terminology. Moreau calls

f the function dual to f.Butin

the special case considered by Birnbaum and Orlicz, he talks about the Young

dual. The inequality (úú) is called Young’s inequality by several authors. This

is OK, but I cannot see any reason for calling f ‘æ

f the Young transformation.

If you want to include a personal name, it seems Legendre is the closest at

hand, even though it does not create the correct association. The “maximum

transformation”

I have not seen. That is a possibility. I do not want to add

a new name, but nevertheless say, that if I now should give the transformation

one, I would let me be guided by the analogy and the connection with polarity

between convex sets (in dual vector spaces) and for instance call it parabolic

polarity.

With kind regards

Werner Fenchel

Â — Ê

I thanked Fenchel in a letter dated 1977 April 02 and told him that Christer Borell

and several other persons in Uppsala had discussed it, and that we liked very much

the term parabolic polarity,which I had not seen before. However, I remarked that

it is rather unwieldy to call

f the function parabolically polar to f .

I later met Werner Fenchel at a conference in Uppsala and could continue the

discussion on the history of duality and possible terms.

11 A plethora of terms

As Fenchel wrote concerning names: “I admit that the situation is rather chaotic.”

Since the Fenchel transformation is also an example of a Galois connection, where

the situation concerning concepts and the terms used for them is even more chaotic,

this story can be continued: terms like residual mapping, adjunction, upper adjoint,

and lower adjoint occur. (See my paper 2010, Section 4). The reason for this state

of aﬀairs is pretty clear: the concepts are indeed fundamental in nature, but they

are approached from many diﬀerent directions, often by researchers who are not

aware of earlier work.

Mentioned in my earlier letter to Fenchel; the term was used by Bellman & Karush (1963).

Normat 2/2013 Christer Oscar Kiselman 143

Werner Fenchel in the late sixties or early seventies

In Section 6 above I have used the terms that I prefer now. However, several terms

have been used for the main concepts involved here: for the supporting function

of a set; for a Fenchel transform

f; for the Fenchel transformation f ‘æ

f; and for

Fenchel’s inequality › · x 6 f(x)+

f(›). I list a few of them below.

• le Gendre [Legendre] (1789; Lû le 1

Septembre 1787): “Au lieu de considérer

z, p, q comme des fonctions de x & y, rien n’empêche de regarder x, y, z

comme des fonctions de p & q; [. . . ]” (p. 315). This is the very beginning of

what we now call the Legendre transformation. Here it is about a function z

of two variables x and y with p = ˆz/ˆx and q = ˆz/ˆy. The symmetry or

duality between z, p, q and x, y, z is evident. Later in the article the author

considers functions of m ore than two variables.

• Minkowski 1896: in Minkowski (1910) app ear the terms “Stützebene” (p. 13);

“nirgends concave Fläche” (p. 35), meaning a convex surface; “überall convexe

Fläche” (p. 38), meaning a strictly convex surface. The part containing these

terms was published already in 1896.

• Minkowski 1897: What is now known as “Minkowski addition” appears for

instance in a paper published in 1897, reproduced in Minkowski (1911a:108).

• Minkowski (1903): “konvexer Körper” (p. 447); “Stützebenfunktion” (p. 448).

• Minkowski (1911b): “konvexer Körper” (p. 131); “Stützebene” (p. 136).

• Birnbaum & Orlicz (1931): “konjugierte Potenzen” (p. 2); “Die N-Funktion

N(v) heißt konjugiert” (p. 6); “komplementäre Funktion” (p. 8).

• Bonnesen & Fenchel (1934): “Die Stützfunktion” (p. 23).

144 Christer Oscar Kiselman Normat 2/2013

• Mandelbrojt (1939): “qu’on peut appeler la fonction convexe associée à f(x)”

(p. 977).

• Fenchel (1949): The two functions are called “conjugate” to each other.

• Goldstein (1950): “the Legendre transformation” (p. 215).

• Landau & Lifshitz (1960): “Legendre’s transformation” (p. 131).

• Moreau (1962): “la fonction convexe duale de f ” (p. 2897).

• Bellman & Karush (1963): “maximum convolution” (p. 67); “maximum trans-

form” (p. 68).

• Hörmander (1963): “Legendretransformen (konjugerade funktionen)” (p. 2.4).

• Valentine (1963): “support functional” (p. 473).

• Moreau (1966–1967): “fonction polaire de h” (p. 33); “une paire de fonctions

duales (ou que chacune est la duale de l’autre) si chacune est la fonction

polaire de l’autre” (p. 45).

• Abraham & Marsden (1967): “a Legendre transformation” (p. 122).

• Ioﬀe & Tihomirov (1968): “konvoljucija” (for inﬁmal convolution); “preobra-

zovanie JUnga ili sopryazhennaya”‘Young’s transformation or conjugate’;

“neravenstvo JUnga”‘Young’s inequality’ (p. 55).

• Rockafellar (1970): “This f

is called the conjugate of f” (p. 104); “sup-

port function” (p. 112); “subgradient” (p. 214); “subdiﬀerential” (p. 215); “The

classical Legendre transformation for diﬀerentiable functions deﬁnes a corre-

spondence which, for convex functions, is intimately connected with the con-

jugacy correspondence” (p. 251); “The Legendre conjugate,” “the Legendre

transformation” (p. 256).

• Kiselman (1978): “The Legendre transformation.”

• Desloge (1982): “Legendre transformation” (p. 920).

• Fenchel (1983): “support function” (p. 127).

• Hörmander (1983): “supporting function” (p. 105); “The quadratic forms

[. . . ] are said to be dual” (p. 206); “dual cone” (p. 257).

• Moreau (1993, in a private letter to Hiriart-Urruty): “C’est moi qui ai intro-

duit le terme sous-gradient (et sa traduction anglaise subgradient).”

• Hiriart-Urruty & Lemaréchal (1993): “Theory of Conjugate Functions”

(p. 36); “the so-called Fenchel correspondence, and is closely related to the

Legendre transform” (p. 38); “Young–Fenchel inequality” (p. 38).

• Hörmander (1994): “Conjugate convex functions (Legendre transforms)”

(p. 16); “The Legendre transform (also called the conjugate function)” (p. 17);

“Then the Legendre transform (= conjugate function = Fenchel transform)”

(p. 67); “supporting function” (p. 69).

Normat 2/2013 Christer Oscar Kiselman 145

• Singer (1997): “subdiﬀerential” (p. 19); “Fenchel–Young inequality” (p. 254);

“Fenchel–Ro ckafellar theorem” (p. 273); “Fenchel–Moreau theorem” (p. 433);

• Gao (2000): “Legendre transformation” (p. 27); “Legendre conjugate transfor-

mation” (p. 27); “Legendre Duality Theorem” (p. 29); “Legendre duality pair”

(p. 30); “Legendre conjugate functions” (p. 30); “Legendre–Fenchel transfor-

mation” (p. 32); “Fenchel transformation” (p. 32); “Fenchel-conjugate func-

tion” (p. 32); “Fenchel–Young inequality” (p. 32); “Fenchel-conjugate” (p. 32);

“Fenchel conjugate pair” (p. 234); “sub-diﬀerential” (p. 236); “sub-gradients”

(p. 236).

• Hiriart-Urruty & Lem aréchal (2001): “Legendre transform” (p. 209) for the

original Legendre transform; “The conjugate of a function”; “conjugacy op-

eration”; “the Legendre–Fenchel transform” (p. 211).

• Ehrenpreis (2003): “Legendre transform (sometimes called Young conjugate)”

(p. 153).

• Murota (2003): “Legendre–Fenchel transform” (p. 10); “Legendre–Fenchel

transformation” (p. 10); “Fenchel duality” (p. 12); “subdiﬀerent ial” (p. 80);

“subgradient” (p. 80); “convex conjugate” (p. 81).

• Kiselman (2015): “The Fenchel transformation”; “the Fenchel t ransform”;

“Fenchel’s inequality” (p. 305).

In 1977, I still called the transformation the Legendre transformation, following my

advisor Lars Hörmander (1931–2012), who, with some emphasis, insisted that the

Fenchel transformation is the same as the Legendre transformation.

I later switched to calling it the Fenchel transformation. The motivation for this

is that I am now of the opinion that the results of Lege ndre b elong to diﬀerential

geometry, whereas Fenchel’s seminal work belongs to the theory of ordered sets

and complete lattices, which has a very diﬀerent ﬂavor.

As Fenchel wrote, the name Legendre does not create the correct association.

Rockafellar (1970:251–260) distingishes between the conjugacy correspondence, the

Legendre conjugate, and the Legendre transformation. As I mentioned in Section

6 above, the Fenchel transformation is a tropicalization of the Fourier or Laplace

transformation and does not invoke diﬀerentiability as the Legendre theory does.

It is indeed an early example of the tropicalization of mathematics, which is now

a very active ﬁeld of research.

The terms transform for

f and transformation for f ‘æ

f are very practical and

follow a well-known pattern: the Fourier transformation, the Laplace transforma-

tion, the Radon transformation, the Fenchel transformation, . . .

12 Why Werner Fenchel had to leave Germany

Many scientists in Germany lost their jobs after the Nazi Machtübernahme in 1933.

Fenchel was one of them. It seems appropriate to describe in some detail this

146 Christer Oscar Kiselman Normat 2/2013

development, which is of interest also because of the general decline of science and

culture in Germany. See also Gordin (2015: Chapter 7, Unspeakable).

As described in my pape r (2017), German mathematics and physics were world

leading in many respects during the nineteenth century and the beginning of the

twentieth century. Scientists in many other countries, including Sweden, wrote in

German, but after 1945, this be came rare in Sweden.

The reasons be hind the transition from German to English in scientiﬁc writing

are often considered to be Germany’s defeat in the Second World War and the

subsequent rise of the USA to the world’s leading country in engineering and natural

science. This opinion needs to be qualiﬁed.

12.1 Science education in Germany 1933–1939

A factor which, according to my opinion, has not been given suﬃcient attention, is

how drastic the loss in German science was already before the start of the Second

World War. This can be illustrated by a few ﬁgures.

In the year 1932, the number of students in mathematics, insurance mathemat-

ics, and physics in Germany was 7,139. Just before the war, in the beginning of

1939, this number had gone down to 1,270, a decrease by 82 per cent in seven

years (Mehrtens 1985:85). In Göttingen, where, as Laurent Schwartz wrote in his

memoirs (1997:79), the greatest scientists of the world resided, the number of math-

ematics students went down from 432 till 37, a decrease by 91 per cent during the

same period.

We can compare with how Harald Bohr (1887–1951) described Göttingen in an

earlier epoch:

While Göttingen was in many ways a provincial town, calm and peaceful, the richest

scientiﬁc life ﬂourished there. A spirit of genuine international brotherhood of a

rare intensity reigned there among the many young mathematicians who came from

nearly all over the world to make a pilgrimage to Göttingen, bound together as they

were by their common interest in and love for their science. (Bohr 1952:xxi–xxii)

The importance of Göttingen as a center is further discussed by Thomas Schøtt:

Göttingen and Bohr’s orientation toward this place is not an exceptional instance.

It is one of a recurring pattern of how the scientiﬁc community ﬁts into the various

national societies according to a center and periphery theory [. . . ] (Schøtt 1979:86)

However, the center of a science is not a ﬁxed point. It can move.

12.2 The law of 1933

Among the decisions causing this disaster in higher education and research was the

Gesetz zur Wiederherstellung des Berufsbeamtentums ‘Law for the Restoration of

the Professional Civil Service’, enacted by the government (not by the Reichstag)

on 1933 April 07, little more than two months after the Machtübernahme, and

signed by Der Reichskanzler Adolf Hitler, Der Reichsminister des Innern Frick,

and Der Reichsminister der Finanzen Graf Schwerin von Krosigk.

As the law was ﬁrst drafted by the Interior Minister Wilhelm Frick (1877–

1946; Reichsminister des Innern 1933–1943), all Beamte nicht arischer Abstam-

mung ‘civil servants of non-Aryan descent’, as well as several other categories, were

to be ﬁred immediately at the Reich, Länder, and Municipal levels of government.

Normat 2/2013 Christer Oscar Kiselman 147

However, the President of Germany, Paul von Hindenburg (1847–1934; Reichs-

präsident 1925–1934) objected to the bill until it had been amended to exclude

three classes of civil servants from the ban.

The cardinal provision in the law is Article 3:

§3

(1) Beamte, die nicht arischer Abstammung sind, sind in den Ruhestand (§§ 8

ﬀ.) zu versetzen; soweit es sich um Ehrenbeamte handelt, sind sie aus dem Amtsver-

hältnis zu entlassen. (Gesetz zur Wiederherstellung des Berufsbeamtentums)

Civil servants who are not of Aryan descent are to be put into the state of retire-

ment (§§ 8 et seq.); in so far as it is about a commission of trust, they are to be

dismissed from service. (My translation)

The law contained provisions for reducing the pensions of these retired civil ser-

vants.

The s ec ond paragraph of Article 3 contained certain exceptions, imposed by

President Hindenburg. However, the ﬁrst and sixth articles opened for a broader

application:

§1

(1) Zur Wiederherstellung eines nationalen Berufsbeamtentums und zur Verein-

fachung der Verwaltung können Beamte nach Maßgabe der folgenden Bestimmungen

aus dem Amt entlassen werden, auch wenn die nach dem geltenden Recht hierfür

erforderlichen Voraussetzungen nicht vorliegen. (Gesetz zur Wiederherstellung des

Berufsbeamtentums)

For the restoration of a national professional civil service and in order to simplify

administration, civil servants can, in accordance with the following provisions, be

dismissed even if the required legal conditions are not fullﬁlled. (My translation)

§6

Zur Vereinfachung der Verwaltung können Beamte in den Ruhestand versetzt

werden, auch wenn sie noch nicht dienstunfähig sind. [. . . ] (Gesetz zur Wiederher-

stellung des Berufsbeamtentums)

In order to simplify administration, employees can be transferred to the state of

retirement, even if they are not yet unsuitable for service. [. . . ] (My translation)

12.3 Hilbert

David Hilb e rt (1862–1943) was the founder of the theory of Hilbert spaces and

known for having presented in Paris on 1900 August 08 ten of a total of twenty-

three problems that were to keep mathematicians busy for a long time. He was a

professor at Göttingen and is reported to have summed up the situation there soon

after Hitler’s Machtübernahme thus:

Sitting next to the Nazis’ newly appointed minister of education at a banquet, he

was asked, “And how is mathematics in Göttingen now that it has been freed of the

Jewish inﬂuence?”

“Mathematics in Göttingen?” Hilbert replied. “There is really none any more.”

(Reid 1970:205)

Reinhard Siegmund-Schultze (personal communication 2014 August 19) remarks,

however, that there is no documentation showing that Hilbert actually pronounced

these words in the presence of the minister of education.

12.4 Einstein, Weyl, Fenchel, and many others

Albert Einstein (1879–1955) resigned his position at the Preußische Akademie

der Wissenschaften in a letter dated 1933 March 28, before he was ousted (Pais

148 Christer Oscar Kiselman Normat 2/2013

1982:450).

He accepted an oﬀer (one of many) from the Institute for Advanced

Study in Princeton, NJ. Most dismissed scientists could of course not ﬁnd some-

thing similar.

Hermann Weyl (1885–1955) succeeded Hilbert in Göttingen in 1930. He was not

Jewish, but his wife Friederike Bertha Helene Weyl (1893–1948) was. The law said

nothing about non-Aryan descent of spouses, but, as noted above, it opened wider

possibilites based on “Vereinfachung der Verwaltung.” Weyl did not want to leave

Germany and therefore declined an early oﬀer to join the Institute for Advanced

Study, but later changed his mind and accepted a second oﬀer in 1933.

Another victim of this “Vereinfachung” was Michael A. Sadowsky (1902–1967),

who got his doctorate at the Technische Hochschule Berlin in 1927 and later lost his

teaching permit there: “In 1933 Michael Sadowsky’s teaching permit was revoked

because his w ife was of Jewish descent. However, the legal base

for such a decision

was es tablished only in 1937.” (Brüning e t al. 1998:6; see also Siegmund-Schultze

2009:37–38).

Among the mathematicians forced to retire or who were dismissed were Otto

Blumenthal (1876–1944), Edmund Landau (1877–1938),

Felix Bernstein (1878–

1956), Emmy Noether (1882–1935), Max Born (1882–1970), Richard Courant (1888–

1972), and Stefan Be rgman (1895–1977).

Of all these scientists, I came to know one personally: Werner Fenchel.

12.5 Zentralblatt and Mathematical Reviews

A reviewing journal called Zentralblatt für Mathematik und ihre Grenzgebiete was

founded in 1931 by Otto Neugebauer (1899–1990). It provided reviews and ab-

stracts for articles in pure and applied mathematics. In 1938, the publisher,

Springer-Verlag, asked for written assurances that no Jews would act as review-

ers (Jackson 1997:330). A number of reviewers in England and the United States

resigned in protest (Lehmer 1988:265).

Neugebauer left the Zentralblatt in 1938 when the name of a Jewish editor (Levi-

Civita

) was, without further notice, eliminated from the title page. (Mehrtens

1987:217)

The Zentralblatt no longer registered all mathematical publications. Something had

to be done. A new reviewing journal, named Mathematical Reviews, was started in

the United States. It was founded in 1940 by the same Otto Neugebauer, now at

Brown University. (Nowadays most users rely on its online version MathSciNet.)

In spite of Einstein’s own decision to leave, a caricature exhibited at the Einstein Museum in

Bern (a part of the Musée d’histoire de Berne, Bernisches Historisches Museum) shows Einstein

being kicked down a staircase by a boot, mentioning that he mistakingly thought he was a

Prussian: “Der Haus knecht der Deutschen Gesan dtschaft in Brüssel wurde beauftragt, eine n dort

herumlungernden Asiaten von der Wahnvorstellung, er sei ein Preuße, zu heilen.” ‘The house

servant of the German embassy in Brussels got the commission to free an Asian tramp from his

misconceived idea that he was a Prussian.’

“Legal base” . . . of a very special kind.

Harald Bohr, who worked with Landau in Göttingen, wrote: “Landau made a strong im-

pression on everyone who came into contact with him. His baroque sense of humour and his

exceptional vitality, characterized equally his scientiﬁc research and his teaching. His thinking

was amazingly quick and sure and his standards for precision and exactitute of exposition were

absolute and inexorable.” (Bohr 1952:xxii).

Tullio Levi Civita (1873–1941).

Normat 2/2013 Christer Oscar Kiselman 149

12.6 The end of a glorious era

It is clear that the glorious era for mathematics in Germany ended well before the

beginning of the war—and independently of the outcome of the war as well as of

the rise of the United States.

12.7 Hermann Minkowski

In another epoch, pe ople moved to Germany rather than out of it: Hermann

Minkowski, who was born in the Kovno Governorate in the Russian Empire (now

Kaunas, Lithuania), moved as a young boy with his family to Königsberg in 1872

to avoid persecution. He became a brilliant mathematician, an early pioneer of

convexity theory, and later taught in Bonn, Königsberg, and Zürich (where Ein-

stein studied for him in the Eidgenössische Polytechnische Schule), and ﬁnally in

Göttingen.

Acknowledgments

Tom Fenchel, son of Käte Fenchel, née Sperling, and Werner Fenchel, allowed me

to publish a translation of his father’s letter. In addition, he kindly sent me three

photos of his father and allowed them to be published here. And it was Tinne Hoﬀ

Kjeldsen who brought me into contact with Tom!

Reinhard Siegmund-Schultze provided important historical information, and

Seidon Alsaody, Sharon Rider, and Arild Stubhaug made valuable comments to

earlier ve rsions of the manuscript.

Michael D. Gordin encouraged me to write this paper separately from my ear-

lier paper on language choice in doctoral theses presented at Uppsala University

and in the journal Acta Mathematica, and sent me important comments on both

manuscripts.

Staﬀan Rodhe sent me a link to the seminal article by le Gendre [Legendre]

(1789). Jean-Baptiste Hiriart-Urruty sent encouraging comments, drew my at-

tention to the nice result of Shiri Artstein-Avidan and Vitali Milman (2009), and

sent me a copy of a letter from Jean-Jacques Moreau. Juliusz Brzezinski sent me

comments and corrections.

For all this invaluable help I am most grateful.

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