Euclidean projections of a p-convex body

In this paper we study Euclidean projections of a p-convex body in IR. Precisely, we prove that for any integer k satisfying lnn ≤ k ≤ n/2, there exists a projection of rank k with the distance to the Euclidean ball not exceeding Cp(k/ln(1 + nk )) 1/p−1/2, where Cp is an absolute positive constant depending only on p. Moreover, we obtain precise estimates of entropy numbers of identity operator acting between `p and `r spaces for the case 0 < p < r ≤ ∞. This allows us to get a good approximation for the volume of p-convex hull of n points in IR, p < 1, which shows the sharpness of the announced result.