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.

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).

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 → || x + y || < 2.

• A Banach space (V, || ||) is strictly convex if and only if x ≠ y and || x ||=|| y ||=1 → || αx + (1-α)y || < 1 for all 0<α<1

References

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