Geometry of families of random projections of symmetric convex bodies

Abstract.

We study the diameter of a family of random n-dimensional orthogonal projections of an arbitrary symmetric convex body in ${\Bbb R}^N$, and we show that this diameter is larger than or equal to the square of Euclidean distances of random k-dimensional projections of the body (where $k = (1/2 - \varepsilon)n$, for any $\varepsilon > 0$).The drop of dimension is necessary and the formula is in a certain sense optimal.