Identifying Set Inclusion by Projective Positions and Mixed Volumes

TL;DR

This paper explores methods to determine if one convex body is included in another (up to a shift) by analyzing mixed volumes, surface areas, and projective positions, providing theoretical criteria for inclusion.

Contribution

It introduces new criteria for identifying convex set inclusion using mixed volumes, projective transformations, and Minkowski sums, advancing geometric analysis techniques.

Findings

Inclusion can be identified by comparing volumes or surface areas in all projective positions.

Similar criteria are established for Minkowski sums of convex bodies.

The results provide theoretical tools for convex set inclusion detection.

Abstract

We study a few approaches to identify inclusion (up to a shift) between two convex bodies in ${\mathbb R}^n$. To this goal we use mixed volumes and fractional linear maps. We prove that inclusion may be identified by comparing volume or surface area of all projective positions of the sets. We prove similar results for Minkowski sums of the sets.