  $F$-concavity is a generalization of power concavity and, actually, the
largest available generalization of the notion of concavity. We characterize
the $F$-concavities preserved by the Dirichlet heat flow in convex domains on
${\mathbb R}^n$, and complete the study of preservation of concavity properties
by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed
in some recent papers. More precisely:
  (1) we discover hot-concavity, which is the strongest $F$-concavity preserved
by the Dirichlet heat flow;
  (2) we show that log-concavity is the weakest $F$-concavity preserved by the
Dirichlet heat flow; quasi-concavity is also preserved only for $n=1$;
  (3) we prove that if $F$-concavity does not coincide with log-concavity and
it is not stronger than log-concavity and $n\ge 2$, then there exists an
$F$-concave initial datum such that the corresponding solution to the Dirichlet
heat flow is not even quasi-concave, hence losing any reminiscence of
concavity.
  Furthermore, we find a sufficient and necessary condition for $F$-concavity
to be preserved by the Dirichlet heat flow. We also study the preservation of
concavity properties by solutions of the Cauchy--Dirichlet problem for linear
parabolic equations with variable coefficients and for nonlinear parabolic
equations such as semilinear heat equations, the porous medium equation, and
the parabolic $p$-Laplace equation.
