  Let $u$ be a solution of the Fisher-KPP equation $$ \partial_t u=\Delta
u+f(u),\quad t>0,\ x\in\mathbb{R}^N. $$ We address the following question: does
$u$ become locally planar as $t\to+\infty$ ? Namely, does $u(t_n,x_n+\cdot)$
converge locally uniformly, up to subsequences, towards a one-dimensional
function, for any sequence $((t_n,x_n))_{n\in\mathbb{N}}$ in
$(0,+\infty)\times\mathbb{R}^N$ such that $t_n\to+\infty$ as $n\to+\infty$ ?
This question is in the spirit of a conjecture of De Giorgi for stationary
solutions of Allen-Cahn equations. The answer depends on the initial datum
$u_0$ of $u$. It is known to be affirmative when the support of $u_0$ is
bounded or when it lies between two parallel half-spaces. Instead, the answer
is negative when the support of $u_0$ is "V-shaped". We prove here that $u$ is
asymptotically locally planar when the support of $u_0$ is a convex set
(satisfying in addition a uniform interior ball condition), or, more generally,
when it is at finite Hausdorff distance from a convex set. We actually derive
the result under an even more general geometric hypothesis on the support of
$u_0$. We recover in particular the aforementioned results known in the
literature. We further characterize the set of directions in which $u$ is
asymptotically locally planar, and we show that the asymptotic profiles are
monotone. Our results apply in particular when the support of $u_0$ is the
subgraph of a function with vanishing global mean.
