Relative Isoperimetric Inequality for Domains Outside a Convex Set

Given a convex set C R and a set D R C, the inequality is called the relative isoperimetric inequality. We prove this inequality in three cases: i) when C and D are symmetric about n 1 mutually orthogonal vertical hyperplanes and @D \ @C is a graph over a horizontal hyperplane; ii) when @D @C and @D \ @C are graphs over a subset A of a horizontal hyperplane such that A is symmetric about n 1 mutually orthogonal vertical hyperplanes; iii) when C is an n-dimensional ball. Also, if S is a disk type surface of nonpositive Gaussian curvature and @S is connected and concave, it is proved that . These relative isoperimetric inequalities are sharp