Closed-Form Parametric Equation for the Minkowski Sum of $m$ Ellipsoids in $\mathbb{R}^N$ and Associated Volume Bounds

TL;DR

This paper derives an exact closed-form formula for the Minkowski sum of multiple ellipsoids in any dimension, providing new volume bounds and curvature expressions that improve upon classical inequalities.

Contribution

It introduces a novel closed-form parametric expression for the Minkowski sum of ellipsoids and sharper volume bounds than existing inequalities.

Findings

Exact formula for Minkowski sum boundary of ellipsoids

New volume bounds sharper than Brunn-Minkowski inequality

Expressions for principal curvatures of Minkowski sums

Abstract

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn-Minkowski inequality. A reverse isometric inequality for convex bodies is also given.