“Strictly convex space”的意思、由来-开放百科全书

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Strictly convex space

释义

Properties
See also
References

In mathematics, a strictly convex space is a normed vector space (X, || ||) 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 (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at x and y. Strict convexity is somewhere between an inner product space (all inner product spaces being strictly convex) and a general normed space in terms of structure. It also guarantees the uniqueness of a best approximation to an element in X (strictly convex) 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 (which implies strict convexity), then it is also reflexive by Milman-Pettis theorem.

Properties

The following properties are equivalent to strict convexity.

A normed vector space (X, || ||) is strictly convex if and only if x ≠ y and || x || = || y || = 1 together imply that || x + y || < 2.
A normed vector space (X, || ||) 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 normed vector space (X, || ||) 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 (X, || ||) is strictly convex if and only if the modulus of convexity δ for (X, || ||) satisfies δ(2) = 1.

References

{{cite journal

| last = Goebel
| first = Kazimierz
| title = Convexity of balls and fixed-point theorems for mappings with nonexpansive square
| journal = Compositio Mathematica
| volume = 22
| issue = 3
| year = 1970
| pages = 269–274
}}{{Functional analysis}}

2 : Convex analysis|Normed spaces

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Properties

See also

References