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The hidden geometry in the zeroes of the Steiner polynomial. (Spanish) Zbl 1230.52001

A historical survey is presented on the connections and developments of topics like the theory of Brunn-Minkowski, convex bodies, total mean curvature, the geometry lying behind the zeroes of Steiner polynomials. The contents are told in very good way; for details one might consult the appropriate list of references in the paper. Such like the books by Gruber, Hadwiger, Marden, Minkowski, Oda, Schneider, dealing with subjects like convex geometry, lectures on volume, surface and isoperimetrie, geometry of polynomials, theory of convex bodies. Such like the papers by R. J. Gardner [Bull. Am. Math. Soc., New Ser. 39, No. 3, 355–405 (2002; Zbl 1019.26008)], V. Katsnelson [Complex Anal. Oper. Theory 3, No. 1, 147–220 (2009; Zbl 1171.53049)], dealing with Steiner polynomials and its zeroes and with the Brunn-Minkowski inequality. But there is much more to be learned from this nice paper.

MSC:

52-03 History of convex and discrete geometry
01A55 History of mathematics in the 19th century
01A65 Development of contemporary mathematics
01A60 History of mathematics in the 20th century
52A39 Mixed volumes and related topics in convex geometry
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
26C10 Real polynomials: location of zeros
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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