Strictly convex space


In mathematics, a strictly convex space is a normed vector space 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, the segment joining x and y meets ∂B only at x and y. Strict convexity is somewhere between an inner product space and a general normed space in terms of structure. It also guarantees the uniqueness of a best approximation to an element in X 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, then it is also reflexive by Milman-Pettis theorem.

Properties

The following properties are equivalent to strict convexity.