On the planar $L_p$-Gaussian-Minkowski problem for $0 \leq p<1$

TL;DR

This paper characterizes convex bodies with proportional Gaussian surface area measures as centered disks for certain p-values and explores bounds on these bodies under specific density conditions, revealing differences between symmetric and asymmetric cases.

Contribution

It proves that convex bodies with proportional Gaussian surface area measures are centered disks for p in [0,1), and establishes uniform bounds under density constraints, highlighting asymmetry effects.

Findings

Convex bodies are centered disks when Gaussian surface area measure is proportional for p in [0,1).

Uniform bounds on convex bodies are obtained under density conditions for p=0 and p in (0,1).

Counterexample shows uniform bounds do not exist in asymmetric cases for p in (0,1).

Abstract

In this paper, we show that if $L_p$ Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk when $p\in[0,1)$. Moreover, we investigate $C^0$ estimate of the corresponding convex bodies when the density function of their Gaussian surface area measures have the uniform upper and lower bound. We obtain convex bodies' uniform upper and lower bound when $p=0$ in asymmetric situation and $p\in(0,1)$ in symmetric situation. In fact, for $p\in(0,1)$ , there is a counterexample to claim the uniform bound does not exist in asymmetric situation.