Completions of convex families of convex bodies

The paper discusses the existence of a continuous extension of functions that are defined on subsets of ? ⁿ and whose values are convex bodies in ? ⁿ . This problem arose in convex geometry in connection with the notion, recently introduced in algebraic geometry, of convex Newton-Okounkov bodies.     

Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Our main result states that the hyperspace of convex compact subsets of a compact convex subset X in a locally convex space is an absolute retract if and only if X is an absolute retract of weight ?ω₁. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube Iω₁ is homeomorphic to Iω₁. An analogous result is also proved for the cone over Iω₁. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.     

Typical convex curves on convex surfaces

In the sense of Baire categories, most convex curves on a smooth twodimensional closed convex surface are smooth. Moreover, if the set of all closed geodesics has empty interior in the space of all convex curves, then most convex curves are strictly convex.This paper was written during the author's visit at Western Washington University, whose substantial support is acknowledged.     

On convex optimization without convex representation

We consider the convex optimization problem P:min_x {f(x) : x ? K}{{\rm {\bf P}}:{\rm min}_{\rm {\bf x}} \{f({\rm {\bf x}})\,:\,{\rm {\bf x}}\in{\rm {\bf K}}\}} where f is convex continuously differentiable, and K ì \mathbb Rⁿ{{\rm {\bf K}}\subset{\mathbb R}^n} is a compact convex set with representation {x ? \mathbb Rⁿ : g_j(x) 3 0, j = 1,?,m}{\{{\rm {\bf x}}\in{\mathbb R}^n\,:\,g_j({\rm {\bf x}})\geq0, j = 1,\ldots,m\}} for some continuously differentiable functions (g _j ). We discuss the case where the g _j ’s are not all concave (in contrast with convex programming where they all are). In particular, even if the g _j are not concave, we consider the log-barrier function f_m{\phi_\mu} with parameter μ, associated with P, usually defined for concave functions (g _j ). We then show that any limit point of any sequence (x_m) ì K{({\rm {\bf x}}_\mu)\subset{\rm {\bf K}}} of stationary points of f_m, m? 0{\phi_\mu, \mu \to 0} , is a Karush–Kuhn–Tucker point of problem P and a global minimizer of f on K.     

Nondegenerate families of convex cones and convex polytopes

This paper describes relations between convex polytopes and certain families of convex cones in R ⁿ .The purpose is to use known properties of convex cones in order to solve Helly type problems for convex sets in R ⁿ or for spherically convex sets in S _n , the n-dimensional unit sphere. These results are strongly related to Gale diagrams.     

Combinatorial representation and convex dimension of convex geometries

We develop a representation theory for convex geometries and meet distributive lattices in the spirit of Birkhoff's theorem characterizing distributive lattices. The results imply that every convex geometry on a set X has a canonical representation as a poset labelled by elements of X. These results are related to recent work of Korte and Lovász on antimatroids. We also compute the convex dimension of a convex geometry.Supported in part by NSF grant no. DMS-8501948.     

Empty convex polygons in almost convex sets

A finite set of points, in general position in the plane, is almost convex if every triple determines a triangle with at most one point in its interior. For every ℓ ≥ 3, we determine the maximum size of an almost convex set that does not contain the vertex set of an empty convex ℓ-gon. Partially supported by grants T043631 and NK67867 of the Hungarian NFSR (OTKA).     

A convex analysis approach for convex multiplicative programming

Global optimization problems involving the minimization of a product of convex functions on a convex set are addressed in this paper. Elements of convex analysis are used to obtain a suitable representation of the convex multiplicative problem in the outcome space, where its global solution is reduced to the solution of a sequence of quasiconcave minimizations on polytopes. Computational experiments illustrate the performance of the global optimization algorithm proposed.      

Inequalities for convex sequences and nondecreasing convex functions

In this paper, by using Wu–Debnath’s method, we establish inequalities of Hammer–Bullen type for convex sequences and nondecreasing conve     

Packing a convex domain with similar convex domains

Given a convex domain C and a positive integer k, inscribe k nonoverlapping convex domains into C, all of them similar to C. Denote by f(k) the maximal sum of their circumferences. In this paper it is shown, that for C square, parallelogram or triangle (1) the first increase of f(k) after k = l² occurs not later than at k = l² + 2, (2) constructions can be given, where the following lower bounds are attained for f(k) = f(l² + j):

$\text{(}\text{1}\text{c}\text{) ? l +}\text{(j ? 1)}\text{2l}\text{j}\text{odd}\text{, l? 2}$

$\text{? l +}\text{j}\text{2}\text{(l + 1) j}\text{even}\text{, l?2}$

where c denotes the circumference of C.     

On the equipartition of plane convex bodies and convex polygons

In this paper we consider the problem of partitioning a plane compact convex body into equal-area parts, i.e., an equipartition, by means of chords. We prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, (b) A convex n-gon admits no equipartition produced by more than n chords having a common interior point.