A generalization of the isoperimetric inequality in the hyperbolic plane

Abstract

where the curve is given by a smooth immersion f of a circle into E and w I (p) denotes the winding number of f w.r.t, p ~ E. Equality holds if and only if the curve is a circle traversed in the same direction a number of times. (In particular Banchoff and Pohl derived an inequality of this type which applies to compact immersed submanifolds in euclidean spaces of arbitrary dimensions.) (Cf. also [3] for a polygonal version of (2).) In the spherical case (1) can be written more symmetrically, namely