A logarithmic Sobolev inequality for a probability measure 
on 
bounds the entropy of the square of a function
by its gradient energy. In one normalization, it has the form
 |
for every smooth function 
with compact support, where
is a nonnegative constant
and 
is the Euclidean gradient. For a nonnegative
integrable function 
, the entropy functional is
 |
with 
.
A random vector 
in 
is linearly 
-subgaussian for 
when every unit vector 
satisfies 
. This is a uniform subgaussian
random variable condition on the one-dimensional projections.
