Volumetric invariants and operators on random families of Banach spaces
Tom 159 / 2003
Studia Mathematica 159 (2003), 315-335
MSC: Primary 46B20.
DOI: 10.4064/sm159-2-10
Streszczenie
The geometry of random projections of centrally symmetric convex bodies in ${ \mathbb R}^N$ is studied. It is shown that if for such a body $K$ the Euclidean ball $B_2^N$ is the ellipsoid of minimal volume containing it and a random $n$-dimensional projection $B=P_H(K)$ is “far” from $P_H(B_2^N)$ then the (random) body $B$ is as “rigid” as its “distance” to $P_H(B_2^N)$ permits. The result holds for the full range of dimensions $1 \le n \le \lambda N$, for arbitrary $\lambda \in (0,1)$.
