Optimal transport, Euler equations, Mather and DiPerna-Lions theories

1.5. THE (ANISOTROPIC) ISOPERIMETRIC INEQUALITY 23
An application to Cheeger sets
A Cheeger set E for an open subset Ω ⊂ Rn , n ≥ 2, is any minimizer of the variational problem
P (E)
cm(Ω) := inf | E ⊂ Ω, 0 < |E| < ∞ .
|E| m
In order to avoid trivial situations, it is assumed that Ω has finite measure and that the parameter
m satisfies the constraint
n − 1
m > . (1.5.9)
n
An interesting question is how to provide lower bounds on cm(Ω) in terms of geometric properties
of Ω. The basic estimate in this direction is the Cheeger inequality,
|Ω| m−(n−1)/n cm(Ω) ≥ |B| m−(n−1)/n cm(B) , (1.5.10)
where B is the Euclidean unit ball. The bound is sharp, in the sense that equality holds in
(1.5.10) if and only if Ω = x0 + rB for some x0 ∈ R n and r > 0. In [20] we strengthen this lower
bound in terms of the Fraenkel asymmetry of Ω
A(Ω) := inf
x∈Rn
|Ω∆Br(x)|
|E|
: |Br| = |E| ,
Theorem 1.5.3 Let Ω be an open set in Rn , n ≥ 2, with |Ω| < ∞, and let m satisfy (1.5.9).
Then
|Ω| m−(n−1)/n cm(Ω) ≥ |B| m−(n−1)/n
2
A(Ω)
cm(B) 1 +
,
C(n, m)
where C(n, m) is a constant depending only on n and m. A possible value for C(m, n) is given
by
2
C(n, m) =
+ C0(n),
m − (n − 1)/n
where C0(n) is the constant defined in Theorem 1.5.1.
1.5.2 An isoperimetric-type inequality on constant curvature manifolds
In the case of a Riemannian manifold (M, g), one can try to mimic Gromov’s proof to obtain an
isoperimetric type inequality. However in this case things become extremely more complicated,
since many computations which are trivial on R n involves second derivatives of the distance, and
so in particular Jacobi fields. In [16], in collaboration with Yuxin Ge, we succeeded in adapting
Gromov’s argument to the case of the sphere and the hyperbolic space.
More precisely, let M n (K) denote the n-dimensional simply connected Riemannian manifold
with constant sectional curvature K ∈ R. Set c(x, y) := 1
2 dg(x, y) 2 , where dg(x, y) is the geodesic
distance between x and y on M, and for K ∈ R define
⎧
⎪⎨
GK(r) :=
⎪⎩
( √ Kr) cos( √ Kr)
sin( √ Kr)
if K > 0,
1
if K = 0,
if K < 0,
( √ |K|r) cosh( √ |K|r)
sinh( √ |K|r)