Abstract
Given the convex body E=E(a,b,c) bounded by the ellipsoid with principal axes of lengths 2a, 2b, and 2c, its surface area, S(a,b,c), is a non-elementary integral unless a=b=c, (E is a ball) or two values of a,b, and c are equal (E is a solid spheroid). This leads to upper and lower estimates for S(a,b,c) in terms of the surface areas of balls or spheroids. We derive many of the known inequalities and some new inequalities for the surface areas of ellipsoids using Minkowski sums of ellipsoids and Minkowski's mixed volumes.
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Pfiefer, R.E. Surface area inequalities for ellipsoids using Minkowski sums. Geom Dedicata 28, 171–179 (1988). https://doi.org/10.1007/BF00147449
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DOI: https://doi.org/10.1007/BF00147449