The $L_p$ Chord Minkowski problem in a critical interval

TL;DR

This paper solves the $L_p$ chord Minkowski problem for the range $0 \\leq p < 1$ without symmetry assumptions, expanding the understanding of chord measures and their geometric implications.

Contribution

It provides the first solution to the $L_p$ chord Minkowski problem in the critical interval $0 \\leq p < 1$ without symmetry constraints.

Findings

Solved the $L_p$ chord Minkowski problem for $0 \\leq p < 1$

Extended the theory of chord measures to a new parameter range

Removed symmetry assumptions in the problem setting

Abstract

Chord measures and $L_p$ chord measures were recently introduced by Lutwak-Xi-Yang-Zhang by establishing a variational formula regarding a family of fundamental integral geometric invariants called chord integrals. Prescribing the $L_p$ chord measure is known as the $L_p$ chord Minkowski problem, which includes the $L_p$ Minkowski problem heavily studied in the past 2 decades as special cases. In the current work, we solve the $L_p$ chord Minkowski problem when $0\leq p<1$, without symmetry assumptions.