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## Abstract

This is an overview of the origin and basic ideas of abstract convexity.
HARPEDONAPTAE AND ABSTRACT CONVEXITY
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  and, in particular, to the tropical theorems of Hahn–Banach type (for
instance, ). 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
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 . 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  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  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  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  on the Seidenberg paper :
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
UV(H, E), provided that Uis the H-support set UH
p:= {hH:hp}of
some element pof E.
Alongside the H-convex sets we consider the so-called H-convex elements. An
element pEis 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  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
xX
(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
ϕ1(U) : x7→ sup
yU
hy, xi.
As usual, ,·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 –.
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=k1k2Kı(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) =
VK1VK2for every VX, the nonoblateness of κmeans that
κV:= (VK1VK2)(VK2VK1)
5
is a zero neighborhood for every zero neighborhood VX. Since κVVV,
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+hKı(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 X0X; i.e., X0=
K1K2=K2K1;
(2) the subspace X0is complemented; i.e., there exists a continuous projection
P:XXsuch 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:XE∪ {+∞} 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 KX, put
πE(K) := {TL(X, E) : T k 0 (kK)}.
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 :XE∪ {+∞} be sublinear operators in
general position. If P(x) + Q(x)0for all xXthen there exists a continuous
linear operator T:XEsuch that
Q(x)T x P(x) (xX).
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 .
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., fl(A, E)
if and only if f:AEand the set {f(α) : αA}is order bounded in E.
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) : αA}(fl(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:XEis called
increasing or isotonic if for all x1, x2Xfrom x1x2it 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+:= {xX:x0}and E+:= {eE:e0}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) = {TL(X, E ) : (xX)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 :αA}is
order bounded for every xX. We denote by hAixthe mapping that assigns the
element αx Eto each αA, i.e. hAix:α7→ αx. If Ais weakly order bounded
then hAixl(A, E) for every ﬁxed xX. Consequently, we obtain the linear
operator hAi:Xl(A, E) that acts as hAi:x7→ hAix. Associate with Aone
more operator
pA:x7→ sup{αx :αA}(xX).
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:XEis a sublinear operator and p2:EFis an
increasing sublinear operator. Then
(p2p1) = T◦ h∂p1i:TL+(l(p1, E), F )Tp1∂p2.
Furthermore, if ∂p1= cop(A1)and p2= cop(A2)then
(p2p1)
=T◦ hA1i:TL+(l(A1, E), F )αεA2TA1=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  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 . 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:XE+and a point xin the eﬀective
domain dom(f) := {xX:f(x)<+∞} of f. Given ε0 in the positive cone
E+of E, by the ε-subdiﬀerential of fat xwe mean the set
ε
f(x) := TL(X, E) : (xX)(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:
e1e2e1e2µ(E)e2e1µ(E).
Since
\
εE
εf(x) = [
εµ(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×YE+and f2:Y×ZE+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
paradigm of distant modeling. Boolean valued analysis  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.
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