Compact Widts in Metric Trees

TL;DR

This paper introduces a new concept of n-widths, called Tn-widths, for metric trees by leveraging convex and compact subsets, and explores their properties and relationships with compact widths.

Contribution

It defines Tn-widths for metric trees, extending the classical n-width concept to a non-linear setting, and analyzes their properties and interrelations.

Findings

Tn-widths are well-defined for metric trees.

The compact width is attained within this framework.

A relationship between compact widths and Tn-widths is established.

Abstract

The definition of $n$-width of a bounded subset $A$ in a normed linear space $X$ is based on the existence of $n$-dimensional subspaces. Although the concept of an $n$-dimensional subspace is not available for metric trees, in this paper, using the properties of convex and compact subsets, we present a notion of $n$-widths for a metric tree, called T$n$-widths. Later we discuss properties of T$n$-widths, and show that the compact width is attained. A relationship between the compact widths and T$n$-widths is also obtained.