Strictly convex space

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.

Properties

• A Banach space (V, || ||) is strictly convex if and only if the modulus of convexity δ for (V, || ||) satisfies δ(2) = 1.
• A Banach space (V, || ||) is strictly convex if and only if x ≠ y and || x || = || y || = 1 together imply that || x + y || < 2.
• A Banach space (V, || ||) is strictly convex if and only if x ≠ y and || x || = || y || = 1 together imply that || αx + (1 − α)y || < 1 for all 0 < α < 1.
• A Banach space (V, || ||) 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.

References

• Goebel, Kazimierz (1970). "Convexity of balls and fixed-point theorems for mappings with nonexpansive square". Compositio Mathematica 22 (3): 269–274.