On a Generalization of the Hermite–Hadamard Inequality and Applications in Convex Geometry

In this paper, we show the following result: if C is an n-dimensional 0-symmetric convex compact set, f : C → [ 0 , ∞ ) is concave, and ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) is not identically zero, convex, with ϕ ( 0 ) = 0 , then 1 | C | ∫ C ϕ ( f ( x ) ) d x ≤ 1 2 ∫ - 1 1 ϕ ( f ( 0 ) ( 1 + t ) ) d t , where | C | denotes the volume of C . If ϕ is strictly convex, equality holds if and only if f is affine, C is a generalized symmetric cylinder and f becomes 0 at one of the basis of C . We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the volume of a central section of it. Second, we also derive a corresponding estimate for log-concave functions.