The norm of the centered Hardy--Littlewood maximal operator

We determine the exact $L^p(\mathbb{R})$ norm of the centered Hardy--Littlewood maximal operator for every $1<p<\infty$, proving that it equals $p/(p-1)$. This settles the sharp strong-type problem for the centered maximal operator on the real line. For every $p$ in this range, the norm is strictly