Mazur spaces.
Properties of metrically convex functions in normed spaces (of any dimension) are considered. The main result, Theorem 4.2, gives necessary and sufficient conditions for a function to be metrically convex, expressed in terms of the classical convexity theory.
We study various aspects of nonexpansive retracts and retractions in certain Banach and metric spaces, with special emphasis on the compact nonexpansive envelope property.
Let , be metric spaces and an injective mapping. We put , and (the distortion of the mapping ). We investigate the minimum dimension such that every -point metric space can be embedded into the space with a prescribed distortion . We obtain that this is possible for , where is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into are obtained by a similar method.
We first introduce a notion of (a,b,c,d)-orthogonality in a normed linear space, which is a natural generalization of the classical isosceles and Pythagorean orthogonalities, and well known α- and (α,β)-orthogonalities. Then we characterize inner product spaces in several ways, among others, in terms of one orthogonality implying another orthogonality.
We show that the Hilbert space is coarsely embeddable into any for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into are equivalent for 1 ≤ p < 2.