A Banach-Stone type theorem for invariant metric groups

Given an invariant metric group (X,d), we prove that the set Lip+1(X) of all nonnegative and 1-Lipschitz maps on (X,d) endowed with the inf-convolution structure is a monoid which completely determines the group completion of (X,d). This gives a Banach-Stone type theorem for the inf-convolution structure in the group framework.