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

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.