Abstract

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 CC be a closed convex cone with nonempty interior such that CC has a bounded section of codimension 11. We show that CC is a cone over an ellipsoid if and only if every bounded section of CC has a center of symmetry. We also show that CC is a cone over an ellipsoid if and only if the affine span of ∂C∩∂(a−C)\partial C \cap \partial(a - C) has codimension 11 for every point aa in the interior of CC. These results generalize the finite-dimensional cases proved by J. Jer{ó}nimo-Castro and T. B. McAllister [Two characterizations of ellipsoidal cones, J. Convex Analysis 20 (2013) 1181–1187].

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Ellipsoidal cone
  • ordered normed linear space
  • centrally symmetric convex body

Mathematics Subject Classification

46B20; 52A50, 46B40, 46B10

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Suggested citation

F. Jafari, T. B. McAllister. “Ellipsoidal Cones in Normed Vector Spaces.” Journal of Convex Analysis 24 (2017), No. 3, 795–805. https://doi.org/10.68381/jca24048

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.