Open, convex, unbounded sets in normed spaces

TL;DR

This paper investigates whether open, convex, unbounded sets in normed spaces can be represented as unions of open straight half lines, establishing this is only true in finite-dimensional spaces.

Contribution

It proves that such a representation holds if and only if the normed space is finite-dimensional, highlighting a fundamental geometric distinction.

Findings

In finite-dimensional spaces, open convex unbounded sets are unions of open half lines.

In infinite-dimensional spaces, this representation does not hold.

The result characterizes the geometric structure of open convex sets in different dimensions.

Abstract

Let X be a normed linear space. We examine if every open, convex and unbounded subset of X is equal to the union of a family of open straight half lines. The answer is affirmative if and only if X is finite dimensional.