Rafael Villa

CONCENTRATION OF THE DISTANCE IN FINITE DIMENSIONAL NORMED SPACES

We prove that in every finite dimensional normed space, for “most” pairs ( x, y ) of points in the unit ball, ║ x − y ║ is more than √2(1 − e). As a consequence, we obtain a result proved by Bourgain, using QS-decomposition, that guarantees an exponentially large number of points in the unit ball any two of which are separated by more than √2(1 − e).

A lower bound for the equilateral number of normed spaces

We show that if the Banach-Mazur distance between an n-dimensional normed space X and l n ∞ is at most 3/2, then there exist n + 1 equidistant points in X. By a well-known result of Alon and Milman, this implies that an arbitrary n-dimensional normed space admits at least e c√log n equidistant points, where c > 0 is an absolute constant. We also show that there exist n equidistant points in spaces sufficiently close to l n p , 1 < p < ∞.

Lower bounds for multilinear forms defined on Hilbert Spaces

In this paper, it is proved that, for any m unit vectors. x 1 …, x m in any n -dimensional real Hilbert space, there exists a unit vector x 0 such that for any y ∈ S n −1 . The exact value of the above integral is calculated, and these results used to improve some lower bounds for multilinear forms on real Hilbert spaces. An integral expression is also given for the complex case.

Maximal Equilateral Sets

A subset of a normed space

An application of mixture distributions in modelization of length of hospital stay.

X

Rogers–Shephard inequality for log-concave functions

X is called equilateral if the distance between any two points is the same. Let

\ell _\infty ^n ({\Bbb C})

m(X)