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Journal of Convex Analysis 20 (2013), No. 4, 1181--1187
Copyright Heldermann Verlag 2013

Two Characterizations of Ellipsoidal Cones

Jesús Jerónimo-Castro
Facultad de Ingenieria, Universidad Autónoma de Querétaro, Cerro de las Campanas s/n, C.P. 76010, Querétaro, México

Tyrrell B. McAllister
Dept. of Mathematics, University of Wyoming, Laramie, WY 82071, U.S.A.


We give two characterizations of cones over ellipsoids. Let $C$ be a closed convex linear cone in a finite-dimensional real vector space. We show that $C$ is a cone over an ellipsoid if and only if the affine span of $\partial C \cap \partial(a - C)$ has dimension $\dim(C) - 1$ for every point $a$ in the relative interior of $C$. We also show that $C$ is a cone over an ellipsoid if and only if every bounded section of $C$ by an affine hyperplane is centrally symmetric.

Keywords: Ellipsoidal cone, centrally symmetric convex body.

MSC: 52A20, 53A07

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