We prove the sharp inequality
J(\Omega):= \frac{\lambda_1(\Omega)} {h_1(\Omega)^2} < \frac{\pi^2}{4},
J
(
Ω
)
:
=
λ
1
(
Ω
)
h
1
(
Ω
)
2
<
π
2
4
,
where
\Omega
Ω
is any planar, convex set,
\lambda_1(\Omega)
λ
1
(
Ω
)
is the first eigenvalue of the Laplacian under Dirichlet boundary conditions, and
h_1(\Omega)
h
1
(
Ω
)
is the Cheeger constant of
\Omega
Ω
. The value on the right-hand side is optimal, and any sequence of convex sets with fixed volume and diameter tending to infinity is a maximizing sequence. Morever, we discuss the minimization of
J
J
in the same class of subsets: we provide a lower bound which improves the generic bound given by Cheeger's inequality, we show the existence of a minimizer, and we give some optimality conditions.