Brunn–Minkowski inequality for multiplicities

This is a convex subset of the real vector space P⊗ZR. In fact, it is a convex polytope; see [Br], where this polytope is discussed from an algebraic point of view. It is known that the total mass of m is a polynomial in m for su ciently large m (denote by k its degree) and m −k m(m · ) weak −→ ( )d ; where d is the Lebesgue measure supported on and the density ( ) is a piecewise-polynomial function; we will not use this piecewise polynomiality in this paper. Recall that a real function f de ned on a convex subset U of a vector space V is called concave, if