# The dual Minkowski problem for negative indices

## TL;DR

This paper provides a complete solution, including existence and uniqueness, for the dual Minkowski problem with negative indices, expanding the understanding of dual curvature measures in convex geometry.

## Contribution

It offers the first comprehensive solution to the dual Minkowski problem for negative indices, covering both existence and uniqueness.

## Key findings

- Established existence of solutions for q < 0
- Proved uniqueness of solutions for q < 0
- Extended the dual Minkowski problem to negative indices

## Abstract

Recently, the duals of Federer's curvature measures, called dual curvature measures, were discovered by Huang, Lutwak, Yang, and Zhang (ACTA, 2016). In the same paper, they posed the dual Minkowski problem, the characterization problem for dual curvature measures, and proved existence results when the index, q, is in (0,n). The dual Minkowski problem includes the Aleksandrov problem (q = 0) and the logarithmic Minkowski problem (q = n) as special cases. In the current work, a complete solution to the dual Minkowski problem whenever q < 0, including both existence and uniqueness, is presented.