Differentiability of quermassintegrals: A classification of convex bodies

Abstract

Abstract. In this paper we characterize the convex bodies in R n whose quermassintegrals satisfy certain differentiability properties, which answers a question posed by Bol in 1943 for the 3-dimensional space. This result will have unexpected consequences on the behavior of the roots of the Steiner polynomial: we prove that there exist many convex bodies in R n , for n ≥ 3, not satisfying the inradius condition in Teissier's problem on the geometric properties of the roots of the Steiner polynomial.

Differentiability properties of the family of p-parallel bodies

“…Further results and applications of the differentiability of quermassintegrals with respect to the one-parameter family of 1-parallel bodies can be found in [10] and the references therein.…”

“…In [6], Hadwiger addressed a closely related question, providing some partial solutions to it. This last question was posed and studied for a general gauge body E and arbitrary dimension n in [10], where the original problem was solved. In this section we study differentiability properties of the functions W i (λ).…”

The role of the form body in bounds for inclusion measures

“…Inner parallel bodies and their properties have been studied in a series of papers (see e.g., [, , , , , , , ]). The study of inner parallel bodies is not only interesting but useful, since it is connected with other nice problems for compact convex sets.…”

“…We should notice that the dimension of ∼ ( ; ) is strictly less than (see [6], [26]). Inner parallel bodies and their properties have been studied in a series of papers (see e.g., [4,8,9,13,[14][15][16][17][18]22,24,32]). The study of inner parallel bodies is not only interesting but useful, since it is connected with other nice problems for compact convex sets.…”

Steiner Polynomials via Ultra-Logconcave Sequences

“…By looking at the roots of particular truncated polynomials, we get rid of the gap, showing that for n = 10, 11 Steiner polynomials are also not weakly stable. Figure 1 depicts the above results, and for further information on the roots of Steiner polynomials in the context of Teissier's problem we refer to [7,9,11,12,13].…”