Minkowski Sum of Ellipsoids and Means of Covariance Matrices

The Minkowski sum and difference of two ellipsoidal sets are in general not ellipsoidal. However, in many applications, it is required to compute the ellipsoidal set which approximates the Minkowski operations in a certain sense. In this study, an approach based on the so-called ellipsoidal calculus, which provides parameterized families of external and internal ellipsoids that tightly approximate the Minkowski sum and difference of ellipsoids, is considered. Approximations are tight along a direction l in the sense that the support functions on l of the ellipsoids are equal to the support function on l of the sum and difference. External (resp. internal) support function-based approximation can be then selected according to minimal (resp. maximal) measures of volume or trace of the corresponding ellipsoid. The connection between the volume-based approximations to the Minkowski sum and difference of two positive definite matrices and their mean using their Euclidean or Riemannian geometries is developed, which is also related to their Bures-Wasserstein mean.