A non-existence result for 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, Böröczky, Colesanti, Yang, The $L_p$-Minkowski problem for $-n<p<1$, Adv. Math. (2019)].

10 pages

References in corpus (1)

• On the planar Lp-Minkowski problem