Strictly convex space

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" || &lt; 1 for all 0 &lt; "α" &lt; 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

* 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&ndash;274


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