Centroid bodies and the convexity of area functionals

TL;DR

This paper introduces a novel volume concept in normed spaces, demonstrating the convexity of associated k-area functionals, including Busemann's 2-volume density, and explores their relation to centroid bodies and affine inequalities.

Contribution

It presents a new volume definition that ensures convexity of k-area functionals and connects these to centroid bodies and affine isoperimetric inequalities.

Findings

k-area functionals are convex for all k

Busemann's 2-volume density is convex

new volume relates to centroid bodies and affine inequalities

Abstract

We introduce a new volume definition on normed vector spaces. We show that the induced $k$-area functionals are convex for all $k$. In the particular case $k=2$, our theorem implies that Busemann's 2-volume density is convex, which was recently shown by Burago-Ivanov. We also show how the new volume definition is related to the centroid body and prove some affine isoperimetric inequalities.