Corpus ID: 237503349

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

@inproceedings{Saroglou2021ANR,
title={A non-existence result for the \$L\_p\$-Minkowski problem},
author={Christos Saroglou},
year={2021}
}
We show that given a real number p < 1, a positive integer n and a proper subspace H of R, the measure on the Euclidean sphere S, which is concentrated in H and whose restriction to the class of Borel subsets of S ∩H equals the spherical Lebesgue measure on S ∩ H , is not the Lp-surface area measure of any convex body. This, in particular, disproves a conjecture from [Bianchi, Böröczky, Colesanti, Yang, The Lp-Minkowski problem for −n < p < 1, Adv. Math. (2019)].

#### References

SHOWING 1-10 OF 24 REFERENCES
The L-Minkowski problem for −n < p < 1
• Mathematics
• 2019
Abstract Chou and Wang's existence result for the L p -Minkowski problem on S n − 1 for p ∈ ( − n , 1 ) and an absolutely continuous measure is discussed and extended to more general measures. InExpand
The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry
• Mathematics
• 2006
Abstract The L p -Minkowski problem introduced by Lutwak is solved for p ⩾ n + 1 in the smooth category. The relevant Monge–Ampere equation (0.1) is solved for all p > 1 . The same equation for p 1Expand
Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems
• Mathematics
• 2016
A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally toExpand
The planar $L_p$-Minkowski problem for $0< p<1$
• Mathematics
• 2016
Necessary and sufficient conditions for the existence of solutions to the asymmetric $L_p$ Minkowski problem in $\mathbb{R}^2$ are established for $0 < p < 1$.
The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem
The Brunn-Minkowski theory is the heart of quantitative convexity. It had its origins in Minkowski's joining his notion of mixed volumes with the Brunn-Minkowski inequality. One of Minkowski's majorExpand
Asymptotic behavior of flows by powers of the Gaussian curvature
• Mathematics
• 2016
We consider a one-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $- K^\alpha \nu$, where $\nu$ denotes the outward-pointing unit normal vector and $\alphaExpand The Gauss Image Problem • Mathematics • 2020 The Brunn-Minkowski theory and the dual Brunn-Minkowski theory are two core theories in convex geometric analysis that center on the investigation of global geometric invariants and geometricExpand The logarithmic Minkowski problem • Mathematics • 2012 In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of aExpand On a non-homogeneous version of a problem of Firey We investigate the uniqueness for the Monge-Ampere type equation $$\label{eq-abstract} det(u_{ij}+\delta_{ij}u)_{i,j=1}^{n-1}=G(u),\ \ \ \ \ \ \ (*)$$on$S^{n-1}\$, whereExpand
On the Lp Minkowski Problem for Polytopes
• Mathematics, Computer Science
• Discret. Comput. Geom.
• 2005
Abstract Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data Lp Minkowski problem for all p > 1.