We study the location and the size of the roots of Steiner polynomials of convex bodies in the Minkowski relative geometry. Based on a problem of Teissier on the intersection numbers of Cartier divisors of compact algebraic varieties it was conjectured that these roots have certain geometric properties related to the in- and circumradius of the convex body. We show that the roots of 1-tangential bodies fulfill the conjecture, but we also present convex bodies violating each of the conjectured properties.
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Martin Henk, María A. Hernández Cifre, Notes on the roots of Steiner polynomials. Rev. Mat. Iberoam. 24 (2008), no. 2, pp. 631–644DOI 10.4171/RMI/550