  We investigate the $m$-relative entropy, which stems from the Bregman
divergence, on weighted Riemannian and Finsler manifolds. We prove that the
displacement $K$-convexity of the $m$-relative entropy is equivalent to the
combination of the nonnegativity of the weighted Ricci curvature and the
$K$-convexity of the weight function. We use this to show appropriate variants
of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the
concentration of measures. We also prove that the gradient flow of the
$m$-relative entropy produces a solution to the porous medium equation or the
fast diffusion equation.
