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 ∂C ∩ ∂(a - C) has codimension 1 for every point a in the interior of C. These results generalize the finite-dimensional cases proved in .
|Original language||English (US)|
|Journal||Journal of Convex Analysis|
|State||Published - Jan 1 2017|
- Centrally symmetric convex body
- Ellipsoidal cone
- Ordered normed linear space