Local affine selections of convex multifunctions
Szymon W\k{a}sowicz
TL;DR
This paper investigates the existence of local affine selections for convex multifunctions, demonstrating that while such selections exist in finite dimensions, they do not always exist in certain infinite-dimensional Banach spaces.
Contribution
The paper provides the first examples of convex multifunctions in infinite-dimensional Banach spaces lacking local affine selections.
Findings
Local affine selections exist in finite-dimensional cases.
Counterexamples show non-existence in specific infinite-dimensional Banach spaces.
Abstract
It is well known that not every convex multifunction admits an affine selection. One could ask whether there exists at least local affine selection. The answer is positive in the finite-dimensional case. The main part of this note consists of two examples of non-existence of local affine selections of convex multifunctions defined on certain infinite-dimensional Banach spaces.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Functional Equations Stability Results