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For a convex set (K) of the (n)-dimensional Euclidean space, theSteiner-Minkowski polynomial (M_K(t)) is defined as the (n)-dimensionalEuclidean volume of the neighborhood of the radius (t). Being defined forpositive (t), the Steiner-Minkowski polynomials are considered for all complex(t). The renormalization procedure for Steiner polynomial is proposed. Therenormalized Steiner-Minkowski polynomials corresponding to all possible solidconvex sets in all dimensions form a normal family in the whole complex plane.For each of the four families of convex sets: the Euclidean balls, the cubes,the regular cross-polytopes and the regular symplexes of dimensions (n), thelimiting entire functions, as (n) tends to infinity, are calculated explicitly.

Steiner-Minkowski Polynomials of Convex Sets in High Dimension, and Limit Entire Functions