On dentability in locally convex vector spaces

TL;DR

This paper establishes a geometric equivalence between V-dentability and V-f-dentability for bounded subsets in locally convex vector spaces, providing a positive answer to a 1978 hypothesis in the case of complete convex bounded metrizable sets.

Contribution

It proves that for a broad class of bounded sets in locally convex spaces, V-dentability and V-f-dentability are equivalent, solving a longstanding hypothesis.

Findings

Equivalence of V-dentability and V-f-dentability for certain bounded sets

Purely geometric proof independent of prior facts

Positive solution to a 1978 hypothesis in specific cases

Abstract

For a locally convex vector space (l.c.v.s.) $E$ and an absolutely convex neighborhood $V$ of zero, a bounded subset $A$ of $E$ is said to be $V$-dentable (respectively, $V$-f-dentable) if for any $\epsilon>0$ there exists an $x\in A$ so that $$x\notin \bar{co} (A\setminus (x+\epsilon V)) $$ (respectively, so that $$ x\notin {co} (A\setminus (x+\epsilon V))). $$ Here, "$\bar{co}$" denotes the closure in $E$ of the convex hull of a set. We present a theorem which says that for a wide class of bounded subsets $B$ of locally convex vector spaces the following is true: $(V)$ every subset of $B$ is $V$-dentable if and only if every subset of $B$ is $V$-f-dentable. The proof is purely geometrical and independent of any related facts. As a consequence (in the particular case where $B$ is complete convex bounded metrizable subset of a l.c.v.s.), we obtain a positive solution to a 1978-hypothesis of Elias Saab (see p. 290 in "On the Radon-Nikodym property in a class of locally convex spaces", Pacific J. Math. 75, No. 1, 1978, 281-291).