On an inequality by Kulikova and Prokhorov

In Kulikova and Prokhorov [Kulikova, A.A., Prokhorov, Yu.V., 2003. Uniform distributions on convex sets: Inequality for characteristic functions. Theory Probab. Appl. 47 (4), 700-701], an upper bound for the characteristic function of uniform distributions on convex sets was established. We prove this inequality in the more general case of absolutely continuous distributions on convex sets with concave densities. In addition, we show that the upper bound given by Kulikova and Prokhorov can be improved.