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We discuss the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals and determinants of symmetric positive matrices respectively. We obtain a local version of the Aleksandrov-Fenchel inequality W 2 i ≥ W i-1 W i+1 which relates the quermassintegrals of a convex body K to those of an arbitrary hyperplane projection of K. A consequence is the following fact: for any convex body K, for any (n - 1)-dimensional subspace E of R n and any t > 0, |P E (K)+tD E |/|P E (K)| ≤ |K+2tD n |/|K|, where D denotes the Euclidean unit ball and |.| denotes volume in the appropriate dimension.

On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body