The location of the hot spot in a grounded convex conductor Lorenzo BrascoRolando MagnaniniPaolo Salani 35K0535B3835B50Heat equationhot spoteigenfunctionsSantalo point We investigate the location of the (unique) hot spot in a convex heat conductor with uniform initial temperature and with boundary grounded at zero temperature. We present two methods to locate the hot spot: the first is based on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and Monge-Amp\` ere equations; the second relies on Alexandrov's reflection principle. Then we show how such a problem can be simplified in case the conductor is a polyhedron. Finally, we present some numerical computations. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4578 10.1512/iumj.2011.60.4578 en Indiana Univ. Math. J. 60 (2011) 633 - 660 state-of-the-art mathematics http://iumj.org/access/