Abstract
Variational mean curvatures were introduced by Massari (Arch Ration Mech Anal 55:357–382, 1974) in the Euclidean setting. Later, Barozzi et al. (Proc AMS 99:313–316, 1987) proved that any set of finite perimeter has a summable variational mean curvature. This result was strongly improved by Barozzi (Rend Mat Acc Lincei S 9(5):149–159, 1994). The main goal of this paper is to show that the construction of variational mean curvatures can be still performed in an abstract measure setting, by axiomatizing a few abstract properties that any perimeter should have. This includes perimeters in suitable metric measure spaces, as introduced in Miranda in (J Math Pure Appl 82:975–1004, 2003) (which includes a variety of situations, as for example, Carnot-Caratheodory spaces), perimeters in spaces with density and nonlocal or fractional perimeters as well.
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