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HARPEDONAPTAE AND ABSTRACT CONVEXITY

S. S. KUTATELADZE

Abstract. This is an overview of the origin and basic ideas of abstract con-

vexity.

The idea of convexity feeds generation, separation, calculus, and approximation.

Generation appears as duality; separation, as optimality; calculus, as representa-

tion; and approximation, as stability. Convexity is traceable from the remote ages

and ﬂourishes in functional analysis.

1. Harpedonaptae

Once upon a time mathematics was everything. It is not now but still carries the

genome of mathesis universalis. Abstraction is the mother of reason and the gist

of mathematics. It enables us to collect the particular instances of any many with

some property we observe or study. Abstraction entails generalization and proceeds

by analogy. The latter lies behind the algebraic approach to idempotent functional

analysis [1] and, in particular, to the tropical theorems of Hahn–Banach type (for

instance, [2]). Analogy is tricky and sometimes misleading. So, it is reasonable to

overview the true origins of any instance of analogy from time to time. This article

deals with abstract convexity which is the modern residence of Hahn–Banach.

Sometimes mathematics resembles linguistics and pays tribute to etymology,

hence, history. Today’s convexity is a centenarian, and abstract convexity is much

younger. Vivid convexity is full of abstraction, but traces back to the idea of a solid

ﬁgure which stems from Euclid.

Sometimes mathematics resembles linguistics and pays tribute to etymology,

hence, history. Today’s convexity is a centenarian, and abstract convexity is much

younger. Vivid convexity is full of abstraction, but traces back to the idea of a solid

ﬁgure which stems from Euclid. Book I of his Elements expounded plane geometry

and deﬁned a boundary and a ﬁgure as follows:

Deﬁnition 13. A boundary is that which is an extremity of anything.

Deﬁnition 14. A ﬁgure is that which is contained by any boundary or boundaries.

Narrating solid geometry in Book XI, Euclid travelled in the opposite direction

from a solid to a surface:

Deﬁnition 1. A solid is that which has length, breadth, and depth.

Deﬁnition 2. An extremity of a solid is a surface.

He proceeded with the relations of similarity and equality for solids:

Date: August 26, 2007.

This article bases on a talk on abstract convexity and cone-vexing abstraction at the Interna-

tional Workshop “Idempotent and Tropical Mathematics and Problems of Mathematical Physics,”

Moscow, August 25–30, 2007.

1

2 S. S. KUTATELADZE

Deﬁnition 9. Similar solid ﬁgures are those contained by similar planes equal in

multitude.

Deﬁnition 10. Equal and similar solid ﬁgures are those contained by similar planes

equal in multitude and magnitude.

These deﬁnitions seem vague, obscure, and even unreasonable if applied to the

ﬁgures other than convex polyhedra. Euclid also introduced a formal concept of

“cone” which has a well-known natural origin. However, convexity was ubiquitous

in his geometry by default. The term “conic sections” was coined as long ago as

200 BCE by Apollonius of Perga. However, it was long before him that Plato

had formulated his famous allegory of cave [5]. The shadows on the wall are often

convex.

Now we can see in Euclid’s implicit deﬁnition of a convex solid body the inter-

section of half-spaces. However, the concept of intersection belongs to set theory

which appeared only at the end of the nineteenth century. We can look at the

Euclid deﬁnition in another fashion and see an analogy with the beautiful and deep

results such as the Cauchy lemma or the Alexandrov solution of the Minkowski

problem. These views are not uncommon of the Euclid approach to the deﬁnition

of a solid ﬁgure. However, it is wiser to seek for the origins of the ideas of Euclid

in his past rather than his future. Euclid was a scientist not a foreteller.

The predecessors of Euclid are the harpedonaptae of Egypt as often sounds at

the lectures on the history of mathematics. The harpedonaptae or rope stretchers

measured tracts of land in the capacity of surveyors. They administered cadastral

surveying which gave rise to the notion of geometry.

If anyone stretches a rope that surrounds however many stakes, he will distin-

guish a convex polyhedron, which is up to inﬁnitesimals a typical compact convex

set or abstract subdiﬀerential of the present-day mathematics. The rope-stretchers

discovered convexity experimentally by measurement. Hence, a few words are in

order about these forefathers of their Hahn–Banach next of kin of today.

Herodotus wrote in Item 109 of Book II Enerpre [3] as follows:

Egypt was cut up: and they said that this king distributed the land to all the

Egyptians, giving an equal square portion to each man, and from this he made his

revenue, having appointed them to pay a certain rent every year: and if the river

should take away anything from any man’s portion, he would come to the king and

declare that which had happened, and the king used to send men to examine and

to ﬁnd out by measurement how much less the piece of land had become, in order

that for the future the man might pay less, in proportion to the rent appointed:

and I think that thus the art of geometry was found out and afterwards came into

Hellas also.

Herodotus never used the term “harpedonaptae” but this word is registered

ﬁrstly in a letter of Democritus (460–370 BCE) who belonged to the same epoch

as Herodotus (484–425 BCE)

Datta in [4] wrote:

. . .One who was well versed in that science was called in ancient India as samkhya-

jna (the expert of numbers), parimanajna (the expert in measuring), sama-sutra-

niranchaka (uniform-rope-stretcher), Shulba-vid (the expert in Shulba) and Shulba-

pariprcchaka (the inquirer into the Shulba).

3

Shulba also written as ´

Sulva or Sulva was in fact the geometry of vedic times

as codiﬁed in ´

Sulva S ¯utras. Since “veda” means knowledge, the vedic epoch and

literature are indispensable for understanding the origin and rise of mathematics.

In 1978 Seidenberg [6] wrote:

Old-Babylonia [1700 BC] got the theorem of Pythagoras from India or that both

Old-Babylonia and India got it from a third source. Now the Sanskrit scholars

do not give me a date so far back as 1700 B.C. Therefore I postulate a pre-Old-

Babylonian (i.e., pre-1700 B.C.) source of the kind of geometric rituals we see

preserved in the Sulvasutras, or at least for the mathematics involved in these

rituals.

Some recent facts and evidence prompt us that the roots of rope-stretching spread

in a much deeper past than we were accustomed to acknowledge. Although the exact

chronology still evades us, the time has come to terminate this digression to the

origins of geometry by the two comments of Kak [7] on the Seidenberg paper [6]:

That was before archaeological ﬁnds disproved the earlier assumption of a break in

Indian civilization in the second millennium B.C.E.; it was this assumption of the

Sanskritists that led Seidenberg to postulate a third earlier source. Now with our

new knowledge, Seidenberg’s conclusion of India being the source of the geometric

and mathematical knowledge of the ancient world ﬁts in with the new chronology

of the texts.

. . . in the absence of conclusive evidence, it is prudent to take the most conserva-

tive of these dates, namely 2000 B.C.E. as the latest period to be associated with

the Rigveda.

Stretching a rope taut between two stakes produces a closed straight line segment

which is the continuum in modern parlance. Rope stretching raised the problem of

measuring the continuum. The continuum hypothesis of set theory is the shadow of

the ancient problem of harpedonaptae. Rope stretching independent of the position

of stakes is uniform with respect to direction in space. The mental experiment of

uniform rope stretching yields a compact convex ﬁgure. The harpedonaptae were

experts in convexity.

We see now that convexity is coeval with the basic ideas of experimental science

and intrinsic to the geometric outlook of Ancient Greece where mathematics had

acquired the ripen beauty of the science and art of provable calculations. It took

millennia to transform sensual perceptions into formal mathematical deﬁnitions.

Convexity has found solid grounds in set theory. The Cantor paradise became

an oﬃcial residence of convexity. Abstraction becomes an axiom of set theory. The

abstraction axiom enables us to reincarnate a property, in other words, to collect

and to comprehend. The union of convexity and abstraction was inevitable. Their

child is abstract convexity.

2. Generation

Let Ebe a complete lattice Ewith the adjoint top >:= +∞and bottom

⊥:= −∞. Unless otherwise stated, Yis usually a Kantorovich space which is a

Dedekind complete vector lattice in another terminology. Assume further that His

some subset of Ewhich is by implication a (convex) cone in E, and so the bottom of

Elies beyond H. A subset Uof His convex relative to Hor H-convex , in symbols

4 S. S. KUTATELADZE

U∈V(H, E), provided that Uis the H-support set UH

p:= {h∈H:h≤p}of

some element pof E.

Alongside the H-convex sets we consider the so-called H-convex elements. An

element p∈Eis H-convex provided that p= sup UH

p; i.e., prepresents the supre-

mum of the H-support set of p. The H-convex elements comprise the cone which

is denoted by C(H, E). We may omit the references to Hwhen His clear from

the context. It is worth noting that convex elements and sets are “glued together”

by the Minkowski duality ϕ:p7→ UH

p. This duality enables us to study convex

elements and sets simultaneously.

Since the classical results by Fenchel [8] and H¨ormander [9, 10] we know deﬁnitely

that the most convenient and conventional classes of convex functions and sets are

C(A(X),RX) and V(X0,RX). Here Xis a locally convex space, X0is the dual

of X, and A(X) is the space of aﬃne functions on X(isomorphic with X0×R).

In the ﬁrst case the Minkowski duality is the mapping f7→ epi(f∗) where

f∗(y) := sup

x∈X

(hy, xi − f(x))

is the Young–Fenchel transform of for the conjugate function of f. In the second

case we prefer to write down the inverse of the Minkowski duality which sends U

in V(X0,RX) to the standard support function

ϕ−1(U) : x7→ sup

y∈U

hy, xi.

As usual, h·,·i stands for the canonical pairing of X0and X.

This idea of abstract convexity lies behind many current objects of analysis and

geometry. Among them we list the “economical” sets with boundary points meeting

the Pareto criterion, capacities, monotone seminorms, various classes of functions

convex in some generalized sense, for instance, the Bauer convexity in Choquet

theory, etc. It is curious that there are ordered vector spaces consisting of the

convex elements with respect to narrow cones with ﬁnite generators. To compute

the meet or join of two reals is nor harder than to compute their sum or product.

This simple observation is one of the underlying ideas of supremal generation and

idempotent analysis. Many diverse aspects of abstract convexity are set forth and

elaborated, for instance, in [11]–[16].

3. Separation

The term “Hahn–Banach” reminds us of geometric separation or algebraic exten-

sion. These hypostases of convexity are omnipresent in modern books on functional

analysis.

Consider cones K1and K2in a topological vector space Xand put κ:=

(K1, K2). Given a pair κdeﬁne the correspondence Φκfrom X2into Xby the

formula

Φκ:= {(k1, k2, x)∈X3:x=k1−k2∈Kı(ı:= 1,2)}.

Clearly, Φκis a cone or, in other words, a conic correspondence.

The pair κis nonoblate whenever Φκis open at the zero. Since Φκ(V) =

V∩K1−V∩K2for every V⊂X, the nonoblateness of κmeans that

κV:= (V∩K1−V∩K2)∩(V∩K2−V∩K1)

5

is a zero neighborhood for every zero neighborhood V⊂X. Since κV⊂V−V,

the nonoblateness of κis equivalent to the fact that the system of sets {κV}serves

as a ﬁlterbase of zero neighborhoods while Vranges over some base of the same

ﬁlter.

Let ∆n:x7→ (x, . . . , x) be the embedding of Xinto the diagonal ∆n(X) of Xn.

A pair of cones κ:= (K1, K2) is nonoblate if and only if λ:= (K1×K2,∆2(X)) is

nonoblate in X2.

Cones K1and K2constitute a nonoblate pair if and only if the conic correspon-

dence Φ ⊂X×X2deﬁned as

Φ := {(h, x1, x2)∈X×X2:xı+h∈Kı(ı:= 1,2)}

is open at the zero. Recall that a convex correspondence Φ from Xinto Yis

open at the zero if and only if the H¨ormander transform of X×Φ and the cone

∆2(X)× {0} × R+constitute a nonoblate pair in X2×Y×R.

Cones K1and K2in a topological vector space Xare in general position provided

that

(1) the algebraic span of K1and K2is some subspace X0⊂X; i.e., X0=

K1−K2=K2−K1;

(2) the subspace X0is complemented; i.e., there exists a continuous projection

P:X→Xsuch that P(X) = X0;

(3) K1and K2constitute a nonoblate pair in X0.

Let σnstand for the rearrangement of coordinates

σn: ((x1, y1),...,(xn, yn)) 7→ ((x1, . . . , xn),(y1, . . . , yn))

which establishes an isomorphism between (X×Y)nand Xn×Yn.

Sublinear operators P1, . . . , Pn:X→E∪ {+∞} are in general position if so

are the cones ∆n(X)×Enand σn(epi(P1)× · · · × epi(Pn)). A similar terminology

applies to convex operators.

Given a cone K⊂X, put

πE(K) := {T∈L(X, E) : T k ≤0 (k∈K)}.

We readily see that πE(K) is a cone in L(X, E ).

Theorem. Let K1, . . . , Knbe cones in a topological vector space Xand let E

be a topological Kantorovich space. If K1, . . . , Knare in general position then

πE(K1∩ · · · ∩ Kn) = πE(K1) + · · · +πE(Kn).

This formula opens a way to various separation results.

Sandwich Theorem. Let P, Q :X→E∪ {+∞} be sublinear operators in

general position. If P(x) + Q(x)≥0for all x∈Xthen there exists a continuous

linear operator T:X→Esuch that

−Q(x)≤T x ≤P(x) (x∈X).

Many eﬀorts were made to abstract these results to a more general algebraic

setting and, primarily, to semigroups. The relevant separation results are collected

in [17].

4. Calculus

Consider a Kantorovich space Eand an arbitrary nonempty set A. Denote by

l∞(A, E) the set of all order bounded mappings from Ainto E; i.e., f∈l∞(A, E)

if and only if f:A→Eand the set {f(α) : α∈A}is order bounded in E.

6 S. S. KUTATELADZE

It is easy to verify that l∞(A, E) becomes a Kantorovich space if endowed with

the coordinatewise algebraic operations and order. The operator εA,E acting from

l∞(A, E) into Eby the rule

εA,E :f7→ sup{f(α) : α∈A}(f∈l∞(A, E))

is called the canonical sublinear operator given Aand E. We often write εAinstead

of εA,E when it is clear from the context what Kantorovich space is meant. The

notation εnis used when the cardinality of Aequals nand we call the operator εn

ﬁnitely-generated.

Let Xand Ebe ordered vector spaces. An operator p:X→Eis called

increasing or isotonic if for all x1, x2∈Xfrom x1≤x2it follows that p(x1)≤p(x2).

An increasing linear operator is also called positive. As usual, the collection of all

positive linear operators in the space L(X, E ) of all linear operators is denoted by

L+(X, E). Obviously, the positivity of a linear operator Tamounts to the inclusion

T(X+)⊂E+, where X+:= {x∈X:x≥0}and E+:= {e∈E:e≥0}are the

positive cones in Xand Erespectively. Observe that every canonical operator is

increasing and sublinear, while every ﬁnitely-generated canonical operator is order

continuous.

Recall that ∂p := ∂p(0) = {T∈L(X, E ) : (∀x∈X)T x ≤p(x)}is the subdiﬀer-

ential at the zero or support set of a sublinear operator p.

Consider a set Aof linear operators acting from a vector space Xinto a Kan-

torovich space E. The set Ais weakly order bounded if the set {αx :α∈A}is

order bounded for every x∈X. We denote by hAixthe mapping that assigns the

element αx ∈Eto each α∈A, i.e. hAix:α7→ αx. If Ais weakly order bounded

then hAix∈l∞(A, E) for every ﬁxed x∈X. Consequently, we obtain the linear

operator hAi:X→l∞(A, E) that acts as hAi:x7→ hAix. Associate with Aone

more operator

pA:x7→ sup{αx :α∈A}(x∈X).

The operator pAis sublinear. The support set ∂pAis denoted by cop(A) and

referred to as the support hull of A. These deﬁnitions entail the following

Theorem. If pis a sublinear operator with ∂p = cop(A)then P=εA◦ hAi.

Assume further that p1:X→Eis a sublinear operator and p2:E→Fis an

increasing sublinear operator. Then

∂(p2◦p1) = T◦ h∂p1i:T∈L+(l∞(∂p1, E), F )∧T◦∆∂p1∈∂p2.

Furthermore, if ∂p1= cop(A1)and ∂ p2= cop(A2)then

∂(p2◦p1)

=T◦ hA1i:T∈L+(l∞(A1, E), F )∧∃α∈∂εA2T◦∆A1=α◦ hA2i.

Hahn–Banach in the classical formulation is of course the simplest chain rule for

removing any linear embedding from the subdiﬀerential sign. More details and a

huge list of references on subdiﬀerential calculus are collected in [18] along with as

some topical applications to optimality.

5. Approximation

Convexity of harpedonaptae was stable in the sense that no variation of stakes

within the surrounding rope can ever spoil the convexity of the tract to be surveyed.

Study of stability in abstract convexity is accomplished sometimes by introduc-

ing various epsilons in appropriate places. One of the earliest excursions in this

7

direction is connected with the classical Hyers–Ulam stability theorem for ε-convex

functions. The most recent results are collected in [19]. Exact calculations with

epsilons and sharp estimates are sometimes bulky and slightly mysterious. Some al-

ternatives are suggested by actual inﬁnities, which is illustrated with the conception

of inﬁnitesimal optimality.

Assume given a convex operator f:X→E∪+∞and a point xin the eﬀective

domain dom(f) := {x∈X:f(x)<+∞} of f. Given ε≥0 in the positive cone

E+of E, by the ε-subdiﬀerential of fat xwe mean the set

∂ε

f(x) := T∈L(X, E) : (∀x∈X)(T x −F x ≤T x −f x +ε),

with L(X, E) standing as usual for the space of linear operators from Xto E.

Distinguish some downward-ﬁltered subset Eof Ethat is composed of positive

elements. Assuming Eand Estandard, deﬁne the monad µ(E) of Eas µ(E) :=

T{[0, ε] : ε∈◦

E}. The members of µ(E) are positive inﬁnitesimals with respect

to E. As usual, ◦

Edenotes the external set of all standard members of E, the

standard part of E.

We will agree that the monad µ(E) is an external cone over ◦Rand, moreover,

µ(E)∩◦

E= 0. In application, Eis usually the ﬁlter of order-units of E. The

relation of inﬁnite proximity or inﬁnite closeness between the members of Eis

introduced as follows:

e1≈e2↔e1−e2∈µ(E)∧e2−e1∈µ(E).

Since

\

ε∈◦E

∂εf(x) = [

ε∈µ(E)

∂εf(x);

therefore, the external set on both sides is the so-called inﬁnitesimal subdiﬀerential

of fat x. We denote this set by Df (x). The elements of Df (x) are inﬁnitesimal

subgradients of fat x. If the zero operator is an inﬁnitesimal subgradient of fat x

then xis called an inﬁnitesimal minimum point of f. We abstain from indicating

Eexplicitly since this leads to no confusion.

Theorem. Let f1:X×Y→E∪+∞and f2:Y×Z→E∪+∞be convex

operators. Suppose that the convolution f2Mf1is inﬁnitesimally exact at some

point (x, y, z); i.e., (f2Mf1)(x, y)≈f1(x, y) + f2(y, z).If, moreover, the convex

sets epi(f1, Z)and epi(X , f2)are in general position then

D(f2Mf1)(x, y) = Df2(y , z)◦Df1(x, y).

6. Apology

The essence of mathematics resides in freedom, and abstraction is the freedom of

generalization. Freedom is the loftiest ideal and idea of man, but it is demanding,

limited, and vexing. So is abstraction. So are its instances in convexity. Abstract

convexity starts with repudiating the heritage of harpedonaptae, which is annoying

but may turn out rewarding.

Freedom of set theory empowered us with the Boolean valued models yielding

various realizations of the continuum with idempotents galore. Many instances of

Hahn–Banach in modules and semimodules are just the descents or Boolean in-

terpretations of their classical analogs. The celebrated Hahn–Banach–Kantorovich

theorem is simple Hahn–Banach in a Boolean disguise. In fact, we know now that

many sets with idempotents are indistinguishable from their standard analogs in the

8 S. S. KUTATELADZE

paradigm of distant modeling. Boolean valued analysis [20] has greatly changed the

appearance of abstract convexity, while demonstrating that many seemingly new

results are just canny interpretations of harpedonaptae or Hahn–Banach.

“Scholastic” diﬀers from “scholar.” Abstraction is limited by taste, tradition,

and common sense. The challenge of abstraction is alike the call of freedom. But

no freedom is exercised in solitude. The holy gift of abstraction coexists with

gratitude and respect to the legacy of our predecessors who collected the gems of

reason and saved them in the treasure-trove of mathematics.

References

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