The inf-convolution as a law of monoid. An analogue to the Banach-Stone theorem.

Abstract : In this article we study the operation of inf-convolution in a new direction. We prove that the inf-convolution gives a monoid structure to the space of convex $k$-Lipschitz and bounded from below real-valued functions on a Banach space $X$. Then we show that the structure of the space $X$ is completely determined by the structure of this monoid by establishing an analogue to the Banach-Stone theorem. Some applications will be given.