Related papers: Michael selections and Castaing representations wi…
Given a strictly positive measure, we characterize inner semicontinuous solid convex-valued mappings for which continuous functions which are selections almost everywhere are selections. This class contains continuous mappings as well as…
The famous Michael selection theorem deals with the characterisation of paracompact spaces by continuous selections of lower semi-continuous mappings in Banach spaces. In this paper, we will discuss several equivalent forms of this theorem,…
Thamrongthanyalak demonstrated a definable version of Michael's selection theorem in d-minimal expansions of the real field. We generalize this result to the case in which the structures are d-minimal expansions of ordered fields $\mathcal…
Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable…
The Mar\'echal topology, also called the Effros-Mar\'echal topology, is a natural topology one can put on the space of all von Neumann subalgebras of a given von Neumann algebra. It is a result of Mar\'echal from 1973 that this topology is…
We prove that Michael's paraconvex-valued selection theorem for paracompact spaces remains true for C'(E)-valued mappings defined on collectionwise normal spaces. Some possible generalisations are also given.
It is shown that max-preserving maps (or join-morphisms) on the positive orthant in Euclidean $n$-space endowed with the component-wise partial order give rise to a semiring. This semiring admits a closure operation for maps that generate…
Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued…
The following selection theorem is established:\\ Let $X$ be a compactum possessing a binary normal subbase $\mathcal S$ for its closed subsets. Then every set-valued $\mathcal S$-continuous map $\Phi\colon Z\to X$ with closed $\mathcal…
The main result states that every convex set-valued function defined on a real interval with compact values in a locally convex space, admits an affine selection. In the case if the target space is a real line and the values are closed real…
Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several…
We show that if $E$ is a closed convex set in $\mathbb C^n$ $(n>1)$ contained in a closed halfspace $H$ such that $E\cap bH$ is nonempty and bounded, then the concave domain $\Omega = \mathbb C^n\setminus E$ contains images of proper…
We demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve…
We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping…
Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial…
We generalize previous results and demonstrate that the Dirac representation theory can be effectively adjusted and applied to continuous or discrete signals of infinite time duration. The role of the identity and projection operators is…
This paper studies ways to represent an ordered topological vector space as a space of continuous functions, extending the classical representation theorems of Kadison and Schaefer. Particular emphasis is put on the class of semisimple…
This paper establishes three minimax theorems for possibly nonconvex functions on Euclidean spaces or on infinite-dimensional Hilbert spaces. The theorems also guarantee the existence of saddle points. As a by-product, a complete solution…
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…
A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that…