Sugeno integral and geometric inequalities

In this work, we prove a Prekopa-Leindler type inequality for the Sugeno integral. More precisely, if 0 < lambda < 1 and h ((1 - lambda)x + lambda y) >= f(x)(1-lambda) g(y)(lambda), for all x, y is an element of R-n, where h, f and g are nonnegative mu-measurable functions on R-n, then f(Rn) hd mu >= (f(Rn) fd mu) Lambda (f(Rn) gd mu), for any concave fuzzy measure mu. Also, we derive a general Brunn-Minkowski inequality (standard form) for any homogeneous quasiconcave fuzzy measure mu on R-n.