On the Geometry of Locally Nonconical Convex Sets |
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Authors: | Glenn C. Shell |
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Affiliation: | (1) Division of Natural Sciences and Mathematics, Lincoln University, 820 Chestnut Street, Jefferson City, MO, 65102-0029, U.S.A. |
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Abstract: | A convex subset Q of a Hausdorff topological vector space is called locally nonconical (LNC) if for every two points x,yQ there is a relative neighborhood U of x in Q such that U+(y-x) Q. A geometric characterization (Theorem 2.2) of closed LNC sets with nonempty interior in a Hilbert space is supplied. It states that any proper line segment ]x,y[ contained in bd(Q), the topological boundary of Q, lies inside a relative neighborhood in bd(Q) composed of parallel line segments. It is shown that one half of this characterization, at least, generalizes to the setting of a locally convex Hausdorff topological vector space (LCHTVS). This leads to the observation that the set ext(Q) of extreme points of any LNC set Q in an LCHTVS is closed. Finally, it is proven that, in the same setting, all LNC sets are uniformly stable and, hence, stable. |
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Keywords: | stable convex set zonoid. |
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