TITLE: Volume of Minkowski sum of a ball and an ellipsoid
QUESTION [6 upvotes]: Is there a simple way to calculate/estimate the volume of Minkowski sum of an n-dimensional unit ball and an n-dimensional ellipsoid? Even a simple ellipsoid like $\frac{x_1^2}{a^2} + x_2^2 + \ldots + x_n^2 = 1$ will do.

REPLY [3 votes]: For more on this, check out "Surface area and other measures of ellipsoids", by one I. Rivin:
[math/0403375] Surface area and other measures of ellipsoids
by I Rivin 
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Surface area and other measures of ellipsoids - Elsevier
by I Rivin - 2007
Advances in Applied Mathematics 39 (2007)