Strictly convex space

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. 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.