Nicola Abatangelo, Enrico Valdinoci
A notion of nonlocal curvature
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ABSTRACT.  We consider a nonlocal (or fractional) curvature 
and we investigate similarities and differences with respect to the classical 
case. In particular, we show that 
the nonlocal mean curvature may be seen as an average of suitable 
nonlocal directional curvatures and there is a natural asymptotic convergence to the classical case. 
Nevertheless, differently from the classical cases, minimal and maximal nonlocal directional curvatures 
are not in general attained at perpendicular directions and in fact one can 
arbitrarily prescribe the set of extremal directions for nonlocal directional curvatures. 
Also the classical directional 
curvatures naturally enjoy some linear 
properties that are lost 
in the nonlocal framework. 
In this sense, nonlocal directional 
curvatures are somewhat intrinsically nonlinear.