Geodesic
Ellipsoidal Cones in Normed Vector Spaces
Journal of convex analysis, Tome 24 (2017) no. 3, pp. 795-805
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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].
Classification : 46B20, 52A50, 46B40, 46B10
Mots-clés : Ellipsoidal cone, ordered normed linear space, centrally symmetric convex body 
@article{JCA_2017_24_3_JCA_2017_24_3_a3,
     author = {F. Jafari and T. B. McAllister},
     title = {Ellipsoidal {Cones} in {Normed} {Vector} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {795--805},
     year = {2017},
     volume = {24},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2017_24_3_JCA_2017_24_3_a3/}
}
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F. Jafari; T. B. McAllister. Ellipsoidal Cones in Normed Vector Spaces. Journal of convex analysis, Tome 24 (2017) no. 3, pp. 795-805. http://geodesic.mathdoc.fr/item/JCA_2017_24_3_JCA_2017_24_3_a3/