Centres of convex sets in \(L^ p\)-metrics (Q1915580)

It is shown that for each convex body \(A\subset \mathbb{R}^n\) there exists a naturally defined family \({\mathcal G}_A\subset C(S^{n- 1})\) such that for every \(g\in {\mathcal G}_A\), and every convex function \(f: \mathbb{R}\to \mathbb{R}\) the mapping \(y\mapsto \int_{S^{n- 1}} f(g(x)- \langle y, x\rangle)d\sigma(x)\) has a minimizer which belongs to \(A\). As an application, approximation of convex bodies by balls with respect to \(L^p\) metrics is discussed.