Lower and Upper Bounds for Chord Power Integrals of Ellipsoids

  • First we discuss different representations of chord power integrals I_p(K) of any order p >= 0 for convex bodies K (with inner points) in the d-dimensional Euclidean space. Second we derive closed-term expressions of I_p(E(a)) for an ellipsoid E(a) with semi-axes a=(a_1,...,a_d) in terms of the support function of this ellipsoid and prove upper and lower bounds expressed by the volume and the mean breadth of E(a), respectively. A further inequality conjectured for convex body in Davy (1984) is proved for ellipsoids. Some remarks on chord power integrals of superellipsoids and simplices round off the topic. In the Appendix we prove a formula for the (d-1)-volume of (d-1)-ellipsoids arising from the intersection of E(a) with a hyperplane. Further, we derive the exact value of the third-order chord power integral of the Wuerfelecktetraeder correcting a wrong result by Emersleben (1962).

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Author:Lothar HeinrichGND
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/2898
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2014-07)
Publishing Institution:Universität Augsburg
Release Date:2014/11/19
Tag:Poisson cylinder process; integral geometry; convex body; support function; mean breadth; isoperimetric inequality; Carleman's inequality
GND-Keyword:Stochastische Geometrie; Integralgeometrie; Poisson-Prozess; Konvexer Körper; Isoperimetrische Ungleichung
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Stochastik und ihre Anwendungen
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand