Maximal volumes of n-dimensional balls in the p-norm

Abstract

We revisit the well-known problem of determining the dimension in which a unit ball has maximal volume. We consider balls with respect to the p-norm with arbitrary radius.

Given a fixed p, we find all radii for which the volume is maximized in dimension n. Conversely, for a fixed radius, we find all values of p for which the volume is maximal in dimension n.

Keywords

• p-norm
• Ball volume
• Gamma function
• Unimodality
• Logarithmic concavity