Uniqueness of completions and related topics

TL;DR

This paper investigates the conditions under which bounded sets in normed linear spaces have unique completions, exploring the relationships between completeness, constant width, and geometric properties.

Contribution

It introduces and analyzes the property of normed spaces where certain sets have unique completions, distinguishing it from related geometric properties.

Findings

Existence of nontrivial segments with unique completions in certain spaces

The property is weaker than all complete sets being balls

The property is stronger than all constant width sets being balls

Abstract

A bounded subset of a normed linear space is said to be (diametrically) complete if it cannot be enlarged without increasing the diameter. A complete super set of a bounded set $K$ having the same diameter as $K$ is called a completion of $K$. In general, a bounded set may have different completions. We study normed linear spaces having the property that there exists a nontrivial segment with a unique completion. It turns out that this property is strictly weaker than the property that each complete set is a ball, and it is strictly stronger than the property that each set of constant width is a ball. Extensions of this property are also discussed.