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.
