A Cauchy-Crofton formula and monotonicity inequalities for the Barbosa-Colares functionals

Abstract

We prove a Cauchy-Crofton type formula for a class of geometric functionals, here denoted by Ar, r = 0, 1, . . . , n − 1, first considered by L. Barbosa and G. Colares ([BC]) and defined over the space of closed hypersurfaces in a complete simply connected n-dimensional space form. Besides giving an integral-geometric interpretation to these functionals, this formula allows us to prove a monotonicity inequality for the functionals, namely, if M1 and M2 are embedded hypersurfaces enclosing convex regions K1 and K2, respectively, with K1 K2, then Ar(M1) Ar(M2) with equality holding if and only if K1 = K2 (and consequently M1 = M2).