Multidirectional mean value inequalities and weak monotonicity

Multidirectional mean value inequalities provide estimates of the difference of the extremal value of a function on a given bounded set and its value at a given point in terms of its (sub)-gradient at some intermediate point. A generalization of such multidirectional mean value inequalities is derived by using new infinitesimal conditions for a weak r-growth of the lower semicontinuous function along approximate trajectories of differential inclusions. This new form of the multidirectional mean value inequality does not rely on the linear structure of the underlying space and removes a traditional assumption of lower boundedness on the function.