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Convexity and the hemisphere. (English) Zbl 0688.58001

An explicit construction is given of continuous bounded nonnegative convex functions which are defined on the product of a small hemisphere with itself and which vanish precisely on the diagonal. It is shown that no such function exists for the strict hemisphere, even if the boundedness requirement is lifted. The generalization to regular geodesic balls is indicated.
Reviewer: W.S.Kendall

MSC:

58C05 Real-valued functions on manifolds
26B25 Convexity of real functions of several variables, generalizations
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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