A non-existence result for the L_p-Minkowski problem

2021/09/14 by Saroglou, Christos
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2109.06545

Abstract

We show that given a real number p<1, a positive integer n and a proper subspace H of ℝⁿ, the measure on the Euclidean sphere \mathbbS^n-1, which is concentrated in H and whose restriction to the class of Borel subsets of \mathbbS^n-1∩ H equals the spherical Lebesgue measure on \mathbbS^n-1∩ 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

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