A non-existence result for the $L_p$-Minkowski problem

TL;DR

The paper proves that certain measures concentrated on subspaces cannot be realized as $L_p$-surface area measures of convex bodies for $p<1$, disproving a prior conjecture in the field.

Contribution

It establishes a non-existence result for the $L_p$-Minkowski problem when $p<1$, specifically for measures supported on subspaces, challenging previous assumptions.

Findings

Certain measures are not $L_p$-surface area measures for $p<1$

Disproves a conjecture from prior literature

Advances understanding of the $L_p$-Minkowski problem

Abstract

We show that given a real number $p<1$, a positive integer $n$ and a proper subspace $H$ of $\mathbb{R}^n$, the measure on the Euclidean sphere $\mathbb{S}^{n-1}$, which is concentrated in $H$ and whose restriction to the class of Borel subsets of $\mathbb{S}^{n-1}\cap H$ equals the spherical Lebesgue measure on $\mathbb{S}^{n-1}\cap H$, is not the $L_p$-surface area measure of any convex body. This, in particular, disproves a conjecture from [Bianchi, B\"or\"oczky, Colesanti, Yang, The $L_p$-Minkowski problem for $-n<p<1$, Adv. Math. (2019)].