The discrete logarithmic Minkowski problem for the electrostatic $\mathfrak{p}$-capacity

TL;DR

This paper solves the discrete logarithmic Minkowski problem for electrostatic $rak{p}$-capacity in the case where the measure's support is in general position, extending the geometric understanding of capacity-related Minkowski problems.

Contribution

It provides a solution to the discrete logarithmic Minkowski problem for $rak{p}$-capacity with support in general position, for $1<rak{p}<n$, a significant extension in geometric analysis.

Findings

Solved the discrete logarithmic Minkowski problem for $rak{p}$-capacity.

Extended the problem's solution to the case where support is in general position.

Contributed to the geometric understanding of electrostatic capacity and Minkowski problems.

Abstract

The Minkowski problem for electrostatic capacity characterizes measures generated by electrostatic capacity, which is a well-known variant of the Minkowski problem. This problem has been generalized to $L_p$ Minkowski problem for $\mathfrak{p}$-capacity. In particular, the logarithmic case $p=0$ relates to cone-volumes and therefore has a geometric significance. In this paper we solve the discrete logarithmic Minkowski problem for $1<\mathfrak{p}<n$ in the case where the support of the given measure is in general position.