Let be a polynomial with only real zeros having , as consecutive zeros. It was proved by P. Erdős and T. Grünwald that if on , then the ratio of the area under the curve to the area of the tangential rectangle does not exceed . The main result of our paper is a multidimensional version of this result. First, we replace the class of polynomials considered by Erdős and Grünwald by the wider class consisting of functions of the form , where is logarithmically concave on , and show that their result holds for all functions in . More generally, we show that if and , then for all , the integral does not exceed . It is this result that is extended to higher dimensions. Our consideration of the class is crucial, since, unlike the narrower one of Erdős and Grünwald, its definition does not involve the distribution of zeros of its elements; besides, the notion of logarithmic concavity makes perfect sense for functions of several variables.