07-19
Notes on the roots of steiner polynomials
by Henk, M.; Hern\xe1ndez Cifre, M. A.
Preprint series: 07-19, Preprints
- MSC:
- 52A20 Convex sets in $n$ dimensions (including convex hypersurfaces), See also {53A07, 53C45}
- 52A39 Mixed volumes and related topics
- 30C15 Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral), {For algebraic theory, See 12D10; for real methods, See 26C10}
Abstract: 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.
Keywords: Steiner polynomial, Teissier\'s problem, tangential bodies, circumradius, inradius
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