Strictly convex space

Inmathematics,astrictly convex spaceis anormed vector space(X,|| ||) for which the closed unitballis a strictlyconvex set.Put another way, a strictly convex space is one for which, given any two distinct pointsxandyon theunit sphere∂B(i.e. theboundaryof the unit ballBofX), the segment joiningxandymeets ∂Bonlyatxandy.Strict convexity is somewhere between aninner product space(all inner product spaces being strictly convex) and a generalnormed spacein terms of structure. It also guarantees the uniqueness of a best approximation to an element inX(strictly convex) out of a convex subspaceY,provided that such an approximation exists.

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).

If the normed spaceXiscompleteand satisfies the slightly stronger property of beinguniformly convex(which implies strict convexity), then it is also reflexive byMilman–Pettis theorem.

Properties

The following properties are equivalent to strict convexity.

• Anormed vector space(X,|| ||) is strictly convex if and only ifx≠yand ||x|| = ||y|| = 1 together imply that ||x+y|| < 2.
• Anormed vector space(X,|| ||) is strictly convex if and only ifx≠yand ||x|| = ||y|| = 1 together imply that ||αx+ (1 −α)y|| < 1 for all 0 <α< 1.
• Anormed vector space(X,|| ||) is strictly convex if and only ifx≠0andy≠0and ||x+y|| = ||x|| + ||y|| together imply thatx=cyfor some constantc > 0;
• Anormed vector space(X,|| ||) is strictly convexif and only ifthemodulus of convexityδfor (X,|| ||) satisfiesδ(2) = 1.

See also

• Uniformly convex space
• Modulus and characteristic of convexity

References

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