Isometric embeddings of snowflakes into finite-dimensional Banach spaces

TL;DR

This paper proves that only finite metric spaces' snowflakes can be isometrically embedded into finite-dimensional Banach spaces, providing bounds for specific cases involving power functions.

Contribution

It establishes a characterization of metric spaces whose snowflakes embed into finite-dimensional spaces, highlighting the finiteness condition and offering bounds for power functions.

Findings

Snowflakes of infinite metric spaces do not embed into finite-dimensional Banach spaces.

Finite metric spaces' snowflakes can embed isometrically, but only if the space is finite.

Provides bounds on the size of spaces for power function snowflakes depending on the exponent and dimension.

Abstract

We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.