Geometry of families of random projections of symmetric convex bodies

We study the diameter of a family of random n-dimensional orthogonal projections of an arbitrary symmetric convex body in 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 - epsilon)n, for any epsilon > 0). The drop of dimension is necessary and the formula is in a certain sense optimal.