On generalizations of Gowers norms and their geometry

TL;DR

This paper introduces a broad class of norms generalizing Gowers and $L_p$ norms, exploring their properties, examples, and hypergraph characterizations, thus extending the understanding of normed spaces in analysis and combinatorics.

Contribution

It defines a new class of norms based on weighted hypergraphs, generalizing Gowers and $L_p$ norms, and characterizes hypergraph pairs associated with these norms.

Findings

Normed spaces share properties with $L_p$ spaces

Examples include $L_p$, trace norms, and Gowers norms

Results on characterizing hypergraph pairs for norms

Abstract

Motivated by the definition of the Gowers uniformity norms, we introduce and study a wide class of norms. Our aim is to establish them as a natural generalization of the $L_p$ norms. We shall prove that these normed spaces share many of the nice properties of the $L_p$ spaces. Some examples of these norms are $L_p$ norms, trace norms $S_p$ when $p$ is an even integer, and Gowers uniformity norms. Every such norm is defined through a pair of weighted hypergraphs. In regard to a question of Laszlo Lovasz, we prove several results in the direction of characterizing all hypergraph pairs that correspond to norms.