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General results on convex bodies are reviewed and used to derive an exactclosed-form parametric formula for the boundary of the geometric (Minkowski)sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this wasdone through iterative algorithms in which each new ellipsoid was added to anellipsoid approximation of the sum of the previous ellipsoids. Here we provideone shot formulas to add $k$ ellipsoids directly with no intermediateapproximations required. This allows us to observe a new degree of freedom inthe family of ellipsoidal bounds on the geometric sum. We demonstrate anapplication of these tools to compute the reachable set of a discrete-timedynamical system.

Generalized Outer Bounds on the Finite Geometric Sum of Ellipsoids