A note on additive convex processes

We deal with the category $\text{P}$ the objects of which are finite-dimensional, real spaces and the morphisms of which are polyhedral convex processes introduced by R. T. Rockafellar [6]. We give some characterizations of additive processes, and we study superpositions of epimorphisms and monomorphisms in the category $\text{P}$.     

A note on convex fuzzy processes

In this note, we first define the concept of convex fuzzy processes. Second, we present some basic properties of convex fuzzy processes and important connection between convex fuzzy processes and their graphs.     

A note on essentially convex orders

In this note, we determine precisely which partially ordered sets (posets) have the property that, whenever they occur as subposets of a larger poset, they occur there convexly, i.e., as convex subposets. As a corollary, we also determine which lattices have the property that, if they occur as sublattices of a finite distributive lattice L, then they also occur as closed intervals in L. Throughout, all sets will be finite.Dedicated to the memory of Ivan RivalReceived May 5, 2003; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

A note on convex renorming and fragmentability

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology.     

A note on the illumination of convex bodies

LetKE ^d be a convex body and letl _r(K) denote the minimum number ofr-dimensional affine subspaces ofE ^d lying outsideK with which it is possible to illuminateK, where 0rd–1. We give a new proof of the theorem thatl _r(K)(d+1)/(r+1) with equality for smoothK.The work was supported by Hung. Nat. Found. for Sci . Research No. 326-0213 and 326-0113.     

A note on the measurability of convex sets

Archiv der Mathematik -     

A note on the weakly convex and convex domination numbers of a torus

The distanced_G(u,v) between two vertices u and v in a connected graph G is the length of the shortest (u,v) path in G. A (u,v) path of length d_G(u,v) is called a (u,v)-geodesic. A set X⊆V is called weakly convex in G if for every two vertices a,b∈X, exists an (a,b)-geodesic, all of whose vertices belong to X. A set X is convex in G if for all a,b∈X all vertices from every (a,b)-geodesic belong to X. The weakly convex domination number of a graph G is the minimum cardinality of a weakly convex dominating set of G, while the convex domination number of a graph G is the minimum cardinality of a convex dominating set of G. In this paper we consider weakly convex and convex domination numbers of tori.     

A note on the condition numbers of convex sets

Mathematical Programming - In this note, we give a counterexample to show that the characterization formula for the condition numbers of a convex set $$C$$ of $$\mathbb R^n$$ given in Coulibaly and...     

A note on three-parameter families and generalized convex functions

The concept of generalized convex functions introduced by Beckenbach [E.F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937) 363–371] is extended to the two-dimensional case. Using three-parameter families, we define generalized convex (midconvex, M-convex) functions and show some continuity properties of them.     

A note on convex ordering for stable stochastic integrals

We establish a convex ordering between stochastic integrals driven by strictly α-stable processes with index α ∈ (1,2). Our approach is based on the forward–backward stochastic calculus for martingales together with a suitable decomposition of stable stochastic integrals.     

A note on the upper bound for disjoint convex partitions

Let n(k, l,m), k ≤ l ≤ m, be the smallest integer such that any finite planar point set which has at least n(k, l,m) points in general position, contains an empty convex k-hole, an empty convex l-hole and an empty convex m-hole, in which the three holes are pairwise disjoint. In this article, we prove that n(4, 4, 5) ≤ 16.     

A note on d-stability of convex programs and limiting Lagrangians

In this paper we give a criterion for d-stability of convex programs and a perturbation result which subsume and generalize some recent results in semi-infinite programming and limiting Lagrangians.     

A note on the strong polynomiality of convex quadratic programming

We prove that a general convex quadratic program (QP) can be reduced to the problem of finding the nearest point on a simplicial cone inO(n ³ +n logL) steps, wheren andL are, respectively, the dimension and the encoding length of QP. The proof is quite simple and uses duality and repeated perturbation. The implication, however, is nontrivial since the problem of finding the nearest point on a simplicial cone has been considered a simpler problem to solve in the practical sense due to its special structure. Also we show that, theoretically, this reduction implies that (i) if an algorithm solves QP in a polynomial number of elementary arithmetic operations that is independent of the encoding length of data in the objective function then it can be used to solve QP in strongly polynomial time, and (ii) ifL is bounded by a first order exponential function ofn then (i) can be stated even in stronger terms: to solve QP in strongly polynomial time, it suffices to find an algorithm running in polynomial time that is independent of the encoding length of the quadratic term matrix or constraint matrix. Finally, based on these results, we propose a conjecture.corresponding author. The research was done when the author was at the Department of IE & OR, University of California at Berkeley, and partially supported by ONR grant N00014-91-j-1241.