  This paper presents the notion of a variation entropy. This concept is an
entropy framework for the gradient of the solution of a conservation law
instead of on the solution itself. It appears that all semi-norms are
admissible variation entropies. This provides insight into the total variation
diminishing property and justifies it from entropy principles. The framework
allows to derive new local variation diminishing properties in the continuous
form. This can facilitate the design of new numerical methods for problems that
contain discontinuities.
