Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T07:25:30.959Z Has data issue: false hasContentIssue false
  • English
  • Français

EXTENSIONS OF TRANSLATION INVARIANT VALUATIONS ON POLYTOPES

Part of: Polytopes and polyhedra General convexity

Published online by Cambridge University Press:  19 August 2014

Wolfram Hinderer ,
Daniel Hug  and
Wolfgang Weil
Wolfram Hinderer
Affiliation:
Robert-Koch-Str. 196, D-73760 Ostfildern, Germany email wolfram@hinderer-lang.de
Daniel Hug
Affiliation:
Karlsruhe Institute of Technology (KIT), Department of Mathematics, D-76128 Karlsruhe, Germany email daniel.hug@kit.edu
Wolfgang Weil
Affiliation:
Karlsruhe Institute of Technology (KIT), Department of Mathematics, D-76128 Karlsruhe, Germany email wolfgang.weil@kit.edu
Get access

Abstract

We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we introduce flag support measures of convex bodies via a local Steiner formula and derive some of the properties of these measures.

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A39: Mixed volumes and related topics
Secondary: 52B45: Dissections and valuations (Hilbert's third problem, etc.) 52B11: $n$-dimensional polytopes
Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 236 - 258
Copyright
Copyright © University College London 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alesker, S., Theory of valuations on manifolds. I: Linear spaces. Israel J. Math. 156 2006, 311–339.CrossRefGoogle Scholar
Alesker, S., Theory of valuations on manifolds, II. Adv. Math. 207 2006, 420–454.Google Scholar
Alesker, S., On extendability by continuity of valuations on convex polytopes. Adv. Math. 255 2014, 352–380.Google Scholar
Ambartzumian, R. V., Combinatorial integral geometry, metrics and zonoids. Acta Appl. Math. 9 1987, 3–28.Google Scholar
Ambartzumian, R. V., Factorization Calculus and Geometric Probability, Cambridge University Press (Cambridge, 1990).Google Scholar
Arbeiter, E. and Zähle, M., Kinematic relations for Hausdorff moment measures in spherical spaces. Math. Nachr. 153 1991, 333–348.CrossRefGoogle Scholar
Bernig, A., Algebraic integral geometry. In Global Differential Geometry (eds Bär, C., Lohkamp, J. and Schwarz, M.), Springer (Berlin-Heidelberg, 2012), 107–145.Google Scholar
Bianchini, M., Hug, D. and Weil, W., Flag measures of convex sets and convex functions. 2014, in preparation.Google Scholar
Edelsbrunner, H., Algorithms in Combinatorial Geometry, Springer (Berlin, 1987).CrossRefGoogle Scholar
Federer, H., Geometric Measure Theory, Springer (Berlin, 1969).Google Scholar
Hinderer, W., Integral representations of projection functions. PhD Thesis, University of Karlsruhe, Karlsruhe, 2002.Google Scholar
Hug, D. and Schneider, R., Local tensor valuations. Geom. Funct. Anal. 2014, doi:10.1007/s00039-014-0289-0.Google Scholar
Hug, D., Türk, I. and Weil, W., Flag measures for convex bodies. In Asymptotic Geometric Analysis (Fields Institute Communications 68) (eds Ludwig, M., Milman, V. D., Pestov, V. and Tomczak-Jaegermann, N.), Springer (New York, 2013), 145–187.Google Scholar
Joswig, M. and Theobald, T., Polyhedral and Algebraic Methods in Computational Geometry, Springer (London, 2013).Google Scholar
Kropp, R., Erweiterte Oberflächenmaße für konvexe Körper. Diploma Thesis, University of Karlsruhe, Karlsruhe, 1990.Google Scholar
McMullen, P., Weakly continuous valuations on convex polytopes. Arch. Math. 41 1983, 555–564.CrossRefGoogle Scholar
McMullen, P., Valuations and dissections. In Handbook of Convex Geometry B (eds Gruber, P. M. and Wills, J. M.), North-Holland (Amsterdam, 1993), 933–988.Google Scholar
McMullen, P. and Schneider, R., Valuations on convex bodies. In Convexity and its Applications (eds Gruber, P. M. and Wills, J. M.), Birkhäuser (Basel, 1983), 170–247.Google Scholar
Schneider, R., Convex Bodies: The Brunn–Minkowski Theory, 2nd edn., Cambridge University Press (Cambridge, 2014).Google Scholar
Schneider, R. and Weil, W., Stochastic and Integral Geometry, Springer (Berlin–Heidelberg, 2008).CrossRefGoogle Scholar
Weil, W., Zufällige Berührung konvexer Körper durch q-dimensionale Ebenen. Results Math. 4 1978, 84–101.Google Scholar
Weil, W., Kinematic integral formulas for convex bodies. In Contributions to Geometry, Proc. Geometry Symp. (Siegen 1978), (eds Tölke, J. and Wills, J. M.), Birkhäuser (Basel, 1979), 60–76.Google Scholar