Strictly convex space
[Image:Vector norms.svg|frame|right|The unit ball in the middle figure is strictly convex, while the other two balls are not (they contain a line segment as part of their boundary).]
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.- A normed vector space is strictly convex if and only if x ≠ y and || x || = || y || = 1 together imply that || x + y || < 2.
- A normed vector space is strictly convex if and only if x ≠ y and || x || = || y || = 1 together imply that || αx + y || < 1 for all 0 < α < 1.
- A normed vector space is strictly convex if and only if x ≠ 0 and y ≠ 0 and || x + y || = || x || + || y || together imply that x = cy for some constant c > 0;
- A normed vector space is strictly convex if and only if the modulus of convexity δ for satisfies δ = 1.