Centres of Convex Sets inLpMetrics Original Research Article

It is shown that for each convex bodyA⊂Rnthere exists a naturally defined family GA⊂C(Sn−1) such that for everyg∈GA, and every convex functionf: R→Rthe mappingy↦∫Sn−1 f(g(x)−⦠y, x⦔) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toLpmetrics is discussed.