Symmetry of convex sets and its applications to the extremal ellipsoids of convex bodies

Abstract

A convex body K in R n has around it a unique circumscribed ellipsoid CE(K) with minimum volume, and within it a unique inscribed ellipsoid IE(K) with maximum volume. The modern theory of these ellipsoids is pioneered by Fritz John in his seminal 1948 paper. This paper has two, related goals. First, we investigate the symmetry properties of a convex body by studying its (affine) automorphism group Aut(K), and relate this group to the automorphism groups of its ellipsoids. We show that if Aut(K) is large enough, then the complexity of determining the ellipsoids CE(K) and IE(K) is greatly reduced, and in some cases, the ellipsoids can be determined explicitly. We then use this technique to compute the extremal ellipsoids associated with some classes of convex bodies that have important applications in convex optimization, namely when the convex body K is the part of a given ellipsoid between two parallel hyperplanes, and when K is a truncated second-order cone or an ellipsoidal cylinder.

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“…Namely, D 1,1 = D 2,2 = √ d by (12) and there will be no negative update-rates or negative cache-rate.…”

“…Our formulation, though, more closely resembles a geometric problem: min − log |D| under linear constraints corresponds to finding a maximum volume ellipsoid inscribed inside a convex body. It is called the (Löwner-)John ellipsoid and was shown in [12] to be unique.…”

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“…This set is called the automorphism group of K. Applying results from [7], we know that, O(K) ⊆ O(ξ(K)). Since K(t) is symmetric about both the x and y axes, the transformation T(x) = −I 2 x, where I 2 is the identity matrix, lies in O(K(t)), and hence in O(ξ(K(t))).…”

“…Furthermore, if T ∈ O(ξ(K(t))) then T(c) = c. Applying this result with the transformation T(x) = −I 2 x, we get c = (0, 0). We know from [7], that T ∈ O(ξ(K(t))) =⇒ P T H(t)P = H(t). The structure of H(t) appears by applying…”

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