Compactly convex sets in linear topological spaces

TL;DR

The paper introduces the concept of compactly convex sets in linear topological spaces, proves that all convex subsets of the plane are compactly convex, and explores conditions under which such sets have metrizable closures.

Contribution

It defines compactly convex sets, characterizes their properties in various spaces, and provides examples distinguishing convex sets in different dimensions.

Findings

All convex subsets of the plane are compactly convex.

There exists a convex set in R^3 that is not compactly convex.

The closure of a compactly convex set is metrizable iff all its compact subsets are metrizable.

Abstract

A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that $[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex subset of the plane is compactly convex. On the other hand, the space $R^3$ contains a convex set that is not compactly convex. Each compactly convex subset $X$ of a linear topological space $L$ has locally compact closure $\bar X$ which is metrizable if and only if each compact subset of $X$ is metrizable.