Compact Widts in Metric Trees

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.

10 pages