In mathematics, a strictly convex space is a normed topological vector space (V, || ||) for which the unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two points x and y in the boundary ∂B of the unit ball B of V, the affine line L(x, y) passing through x and y meets ∂B only at x and y. Strict convexity is somewhere between an inner product space (all inner product spaces are strictly convex) and a general normed space (all strictly convex normed spaces are normed spaces) in terms of structure. It also guarantees the uniqueness of a best approximation to an element in X (strictly convex) out of Y (a subspace of X) if indeed such an approximation exists.
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