A flow approach to the $L_{-2}$ Minkowski problem

TL;DR

This paper investigates the solvability of the $L_{-2}$ Minkowski problem for smooth, positive, $ ext{pi}$-periodic functions on the circle, establishing density of solutions, necessary conditions, and uniqueness results.

Contribution

It introduces a flow approach to analyze the $L_{-2}$ Minkowski problem, proving density of solvable functions, a necessary solvability condition, and solution uniqueness up to affine transformations.

Findings

Density of solvable functions in the space of smooth, positive, $ ext{pi}$-periodic functions.

A necessary condition for the solvability of the even $L_{-2}$ Minkowski problem.

Uniqueness of solutions up to affine linear transformations.

Abstract

We prove that the set of smooth, $\pi$-periodic, positive functions on the unit circle for which the $L_{-2}$ Minkowski problem is solvable is dense in the set of all smooth, $\pi$-periodic, positive functions on the unit circle with respect to the $L^{\infty}$ norm. Furthermore, we obtain a necessary condition on the solvability of the even $ L_{-2}$ Minkowski problem. At the end, we prove uniqueness of the solutions up to an affine linear transformation.