Strictly convex space | Wikipedia audio article
Channel: wikipedia tts
Date: 27/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Strictly_convex_space


00:00:56 Properties
00:01:15 || y ||
00:01:27 || y ||
00:01:42 || x || + || y || together imply that x
00:02:06 1.
00:02:09 See also



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SUMMARY
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In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at x and y.  Strict convexity is somewhere between an inner product space (all inner product spaces being strictly convex) and a general normed space in terms of structure. It also guarantees the uniqueness of a best approximation to an element in X (strictly convex) out of a convex subspace Y, provided that such an approximation exists.
If the normed space X is complete and satisfies the slightly stronger property of being uniformly convex (which implies strict convexity), then it is also reflexive by Milman-Pettis theorem.