Summary: We say that an even continuous function \(H\) on the unit sphere \(S\) in \(R^n\) admits the Blaschke-Levy representation with \(q>0\) if there exists an even function \(b\) in \(L_1(S)\) so that, for every \(x\) in \(S, H^q(x)\) is equal to the integral over \(S\) of the function \(|(x,z)|^qb(z)\). This representation has numerous applications in convex geometry, probability and Banach space theory.
In this paper, we present a simple formula (in terms of the derivatives of \(H\)) for calculating \(b\) out of \(H\). This formula leads to new estimates for the sup-norm of \(b\) that can be used in connection with isometric embeddings of normed spaces in \(L_q\).