Differentiability of quermassintegrals: A classification of convex bodies
HTML articles powered by AMS MathViewer
- by M. A. Hernández Cifre and E. Saorín PDF
- Trans. Amer. Math. Soc. 366 (2014), 591-609 Request permission
Abstract:
In this paper we characterize the convex bodies in $\mathbb {R}^n$ 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 $\mathbb {R}^n$, for $n\geq 3$, not satisfying the inradius condition in Teissier’s problem on the geometric properties of the roots of the Steiner polynomial.References
- G. Bol, Beweis einer Vermutung von H. Minkowski, Abh. Math. Sem. Hansischen Univ. 15 (1943), 37–56 (German). MR 15824, DOI 10.1007/BF02941073
- T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin-New York, 1974 (German). Berichtigter Reprint. MR 0344997
- Alexander Dinghas, Bemerkung zu einer Verschärfung der isoperimetrischen Ungleichung durch H. Hadwiger, Math. Nachr. 1 (1948), 284–286 (German). MR 29205, DOI 10.1002/mana.19480010503
- Alexander Dinghas, Über eine neue isoperimetrische Ungliechung für konvexe Polyeder, Math. Ann. 120 (1949), 533–538 (German). MR 29203, DOI 10.1007/BF01447844
- Alexander Dinghas, Minkowskische Summen und Integrale. Superadditive Mengenfunktionale. Isoperimetrische Ungleichungen, Mémor. Sci. Math., Fasc. 149, Gauthier-Villars, Paris, 1961 (German). MR 0132456
- M. Fradelizi, A. Giannopoulos, and M. Meyer, Some inequalities about mixed volumes, Israel J. Math. 135 (2003), 157–179. MR 1997041, DOI 10.1007/BF02776055
- Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
- H. Hadwiger, Altes und Neues über konvexe Körper, Birkhäuser Verlag, Basel-Stuttgart, 1955 (German). MR 0073220, DOI 10.1007/978-3-0348-6953-9
- H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0102775, DOI 10.1007/978-3-642-94702-5
- Martin Henk and María A. Hernández Cifre, Notes on the roots of Steiner polynomials, Rev. Mat. Iberoam. 24 (2008), no. 2, 631–644. MR 2459207, DOI 10.4171/RMI/550
- Martin Henk and María A. Hernández Cifre, On the location of roots of Steiner polynomials, Bull. Braz. Math. Soc. (N.S.) 42 (2011), no. 1, 153–170. MR 2774179, DOI 10.1007/s00574-011-0008-5
- M. A. Hernández Cifre and E. Saorín, On the roots of the Steiner polynomial of a 3-dimensional convex body, Adv. Geom. 7 (2007), no. 2, 275–294. MR 2314821, DOI 10.1515/ADVGEOM.2007.016
- María A. Hernández Cifre and Eugenia Saorín, On differentiability of quermassintegrals, Forum Math. 22 (2010), no. 1, 115–126. MR 2604366, DOI 10.1515/FORUM.2010.006
- M. A. Hernández Cifre and E. Saorín, On the volume of inner parallel bodies, Adv. Geom. 10 (2010), no. 2, 275–286. MR 2629815, DOI 10.1515/ADVGEOM.2010.004
- M. A. Hernández Cifre and E. Saorín, On inner parallel bodies and quermassintegrals, Israel J. Math. 177 (2010), 29–47. MR 2684412, DOI 10.1007/s11856-010-0037-6
- Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
- G. Matheron, La formule de Steiner pour les érosions, J. Appl. Probability 15 (1978), no. 1, 126–135 (French). MR 493914, DOI 10.2307/3213242
- Jane Rosamund Sangwine-Yager, INNER PARALLEL BODIES AND GEOMETRIC INEQUALITIES, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–University of California, Davis. MR 2627667
- J. R. Sangwine-Yager, Bonnesen-style inequalities for Minkowski relative geometry, Trans. Amer. Math. Soc. 307 (1988), no. 1, 373–382. MR 936821, DOI 10.1090/S0002-9947-1988-0936821-5
- J. R. Sangwine-Yager, Mixed volumes, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 43–71. MR 1242976, DOI 10.1016/B978-0-444-89596-7.50007-9
- Rolf Schneider, On the Aleksandrov-Fenchel inequality, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 132–141. MR 809200, DOI 10.1111/j.1749-6632.1985.tb14547.x
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
- J. Steiner, Über parallele Flächen, Monatsber. Preuss. Akad. Wiss. (1840), 114–118, [Ges. Werke, Vol II (Reimer, Berlin, 1882) 245–308].
- B. Teissier, Bonnesen-type inequalities in algebraic geometry. I. Introduction to the problem, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 85–105. MR 645731
Additional Information
- M. A. Hernández Cifre
- Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain
- Email: mhcifre@um.es
- E. Saorín
- Affiliation: Institut für Algebra und Geometrie, Universität Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany
- ORCID: 0000-0002-1986-9641
- Email: eugenia.saorin@ovgu.de
- Received by editor(s): February 8, 2010
- Received by editor(s) in revised form: July 13, 2010, January 20, 2011, June 21, 2011, and October 17, 2011
- Published electronically: July 24, 2013
- Additional Notes: The authors were supported by Dirección General de Investigación (MICINN) MTM2009-10418 and by “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, Agencia de Ciencia y Tecnología de la Región de Murcia (Plan Regional de Ciencia y Tecnología 2007/2010), 04540/GERM/06
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 591-609
- MSC (2010): Primary 52A20, 52A39; Secondary 52A40
- DOI: https://doi.org/10.1090/S0002-9947-2013-05723-6
- MathSciNet review: 3130309