For each natural number
n
n
and any bounded, convex domain
Ω
⊂
R
n
\Omega \subset \mathbb {R}^n
we characterize the sharp constant
C
(
n
,
Ω
)
C(n,\Omega )
in the Poincaré inequality
‖
f
−
f
¯
Ω
‖
L
∞
(
Ω
;
R
)
≤
C
(
n
,
Ω
)
‖
∇
f
‖
L
∞
(
Ω
;
R
)
\| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega ) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})}
. Here,
f
¯
Ω
\bar {f}_{\Omega }
denotes the mean value of
f
f
over
Ω
\Omega
. In the case that
Ω
\Omega
is a ball
B
r
B_r
of radius
r
r
in
R
n
\mathbb {R}^n
, we calculate
C
(
n
,
B
r
)
=
C
(
n
)
r
C(n,B_r)=C(n)r
explicitly in terms of
n
n
and a ratio of the volumes of the unit balls in
R
2
n
−
1
\mathbb {R}^{2n-1}
and
R
n
\mathbb {R}^n
. More generally, we prove that
C
(
n
,
B
r
(
Ω
)
)
≤
C
(
n
,
Ω
)
≤
n
n
+
1
d
i
a
m
(
Ω
)
C(n,B_{r(\Omega )}) \leq C(n,\Omega ) \leq \frac {n}{n+1}\mathrm {diam}(\Omega )
, where
B
r
(
Ω
)
B_{r(\Omega )}
is a ball in
R
n
\mathbb {R}^n
with the same
n
−
n-
dimensional Lebesgue measure as
Ω
\Omega
. Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.