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

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.