The Cheeger energy of a function on a metric measure space is the lower semicontinuous relaxation in 
of its squared local Lipschitz slope. For a Lipschitz
function 
,
that slope is 
,
taken as 0 at an isolated point. The energy is
 |
where the infimum is over Lipschitz approximations with finite energy. Its finite-energy
domain is the metric Sobolev
space 
.
On a smooth Riemannian manifold with its volume measure, the energy is
. Quadraticity of
this energy is the Riemannian requirement in the definition of an RCD
space.
