Journal of Convex Analysis 24 (2017), No. 3, 795--805
Copyright Heldermann Verlag 2017
Ellipsoidal Cones in Normed Vector Spaces
Farhad Jafari
Dept. of Mathematics, University of Wyoming, Laramie, WY 82071, U.S.A.
fjafari@uwyo.edu
Tyrrell B. McAllister
Dept. of Mathematics, University of Wyoming, Laramie, WY 82071, U.S.A.
tmcallis@uwyo.edu
[Abstract-pdf]
The characterization of ellipsoids is intimately tied to characterizing the Banach spaces that are Hilbert spaces. We give two characterizations of cones over ellipsoids in real normed vector spaces. Let $C$ be a closed convex cone with nonempty interior such that $C$ has a bounded section of codimension $1$. We show that $C$ is a cone over an ellipsoid if and only if every bounded section of $C$ has a center of symmetry. We also show that $C$ is a cone over an ellipsoid if and only if the affine span of $\partial C \cap \partial(a - C)$ has codimension $1$ for every point $a$ in the interior of $C$. These results generalize the finite-dimensional cases proved by J. Jer{\'o}nimo-Castro and T. B. McAllister [\emph{Two characterizations of ellipsoidal cones}, J. Convex Analysis 20 (2013) 1181--1187].
Keywords: Ellipsoidal cone, ordered normed linear space, centrally symmetric convex body.
MSC: 46B20; 52A50, 46B40, 46B10
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