ON A CHARACTERISTIC PROPERTY OF FINITE-DIMENSIONAL BANACH SPACES

On the Average Distance Property in Finite Dimensional Real Banach Spaces

Approximate Isometries on Finite Dimensional Banach Spaces by Richard

A map T: Ej —► E2 (E|, E2 Banach spaces) is an e-isometry if III T(X) T(Y)\\ \\X Y\\ I < e whenever X, Ye Ex. The problem of uniformly approximating such maps by isometries was first raised by Hyers and Ulam in 1945 and subsequently studied for special infinite dimensional Banach spaces. This question is here broached for the class of finite dimensional Banach spaces. The only positive homogene...

Subspaces of Small Codimension of Finite-dimensional Banach Spaces

Given a finite-dimensional Banach space E and a Euclidean norm on E, we study relations between the norm and the Euclidean norm on subspaces of E of small codimension. Then for an operator taking values in a Hubert space, we deduce an inequality for entropy numbers of the operator and its dual. In this note we study the following problem: given an n-dimensional Banach space E and a Euclidean no...

Isometric Embeddings of Snowflakes into Finite-dimensional Banach Spaces

We consider a general notion of snowflake of a metric space by composing the distance with a nontrivial concave function. We prove that a snowflake of a metric space X isometrically embeds into some finite-dimensional normed space if and only if X is finite. In the case of power functions we give a uniform bound on the cardinality of X depending only on the power exponent and the dimension of t...

A nonconvex separation property in Banach spaces

We establish, in innnite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in optimization, in particular the existence of singular multipliers and show the relation of our principle with the extremal principle of Mordukhovich.