Measures, curvatures and currents in convex geometry

Abstract: This work is devoted to the investigation of basic interrelations between the geometry of convex sets and certain measures (or functionals), curvatures and currents which are associated with such sets.

Section 1 is devoted to the investigation of curvature and surface area measures. The main problem which we address is the exploration of the connection between measure theoretic properties of the curvature and surface area measures and geometric properties of the underlying convex body. The specific measure theoretic property, which we pursue here, is the absolute continuity with respect to a suitable Hausdorff measure. An important technical and geometric device in this context is the notion of generalized curvatures that are defined almost everywhere on the unit normal bundle of a given convex body. In this section, we also establish a stability estimate of optimal order for Minkowski's uniqueness problem, which applies to a large class of convex bodies. Finally, we provide an alternative route to two representations of support measures of convex bodies on sets of sigma-finite Hausdorff measure as weighted Hausdorff measures, where the strength of the singularities of the convex bodies is taken into account. The present approach to one of these theorems is based on a version of Federer's structure theorem in spherical space, for which we give a new integral-geometric proof that is in the spirit of Brian White's recent proof for Federer's structure theorem in Euclidean space.


In Section 2 we extend our framework by replacing the Euclidean unit ball as the fundamental gauge body by a rather arbitrary convex body $B$ containing the origin. In such a setting of relative geometry all measurements and definitions have to refer to the gauge body alone and should essentially be independent of an auxiliary Euclidean structure. Recently, support measures have been introduced in relative geometry for pairs of convex bodies $(K,B)$ such that $o\in B$ and $K$ and $B$ are in general relative position. In particular, if $K$ and $B$ are polytopes in general relative position, then the $B$-support measures of $K$ can be represented in a simple way. Other representations, extending results by Martina Zähle, are obtained if either the support function of $B$ or the support function of $K$ is of class $C^2$. In fact, for a convex body of class $C^2_+$ one can even go one step further and introduce generalized relative curvatures which are defined on the relative normal bundle of $K$ with respect to $B$. On the basis of these new concepts of relative geometry we find another representation for the $B$-support measures which extends a corresponding representation that is known from the Euclidean setting. Subsections 2.3 and 2.4 are devoted to the investigation of relative normals and of certain Euler-type formulae involving relative support measures. The basic question, asking for estimates of the average number of relative normals passing through a point in a convex body, can be traced back to a classical paper by Santal{\'o}. The case of the Minkowski plane deserves to be treated separately, since here stronger results are available due to the surprising fact that the $B$-projection onto $K$ is Lipschitz for all pairs of convex bodies $(K,B)$ in general relative position and for which $o\in\text{int }B$. In the remaining part of the section we treat characterizations of gauge bodies by linear relations between relative support measures, related stability results and a splitting theorem in three-dimensional symmetric Minkowski spaces. These investigations heavily rely on results which are provided in Section 1. Thus Section 2 may be viewed as a general contribution to an overall attempt to transfer concepts and results of Euclidean geometry to general finite-dimensional normed vector spaces.

In Section 3 we combine results about support measures and probabilistic methods of stochastic geometry to gain new information about the structure of contact distributions of random closed sets in the extended convex ring.
Another main subject, discussed in Subsection 3.5, is the investigation of
certain intensity measures which are associated with random measures constructed as non-negative extensions of relative support measures of a random closed set $\Xi$ in the extended convex ring. Our present objective is to establish the absolute continuity of these intensity measures and to determine the explicit form of the densities for general strictly convex gauge bodies. The remaining part of Section 3 is devoted to the derivation of an iterated translative integral formula for relative support measures. Such an integral formula involves certain relative mixed curvature measures which have been of great use in a Euclidean context for applications in stochastic geometry. Now a corresponding tool is available in the framework of relative geometry.

In Section 4 we return to a deterministic and Euclidean setting. Our primary
interest in this section is the investigation of mixed volumes and of the mixed
curvature measures of translative integral geometry. The latter were already treated in Section 3 in the setting of relative geometry, but from a different
point of view and by employing other techniques. These and related mixed
functionals and measures represent a central topic in convex and integral
geometry. The present primary concern is to describe these mixed functionals and measures by using the normal bundles and generalized curvatures of the bodies involved

Angaben zum Objekt

• Standort: Deutsche Nationalbibliothek Frankfurt am Main
• Umfang: Online-Ressource
• Sprache: Englisch
• Anmerkungen: Universität Freiburg, Habilitationsschrift, 1999

Klassifikation und Themen

• Schlagwort: Weights and measures ; Mathematik ; Geometrie ; Konvexe Menge ; Wahrscheinlichkeitstheorie ; Integralgeometrie ; Stochastische Geometrie ; Geometrische Maßtheorie ; Krümmungsmaß ; Krümmung ; Oberflächenmaß

Beteiligte, Orts- und Zeitangaben

• Ereignis: Veröffentlichung
• (wo): Freiburg
• (wer): Universität
• (wann): 2020
• Urheber: Hug, Daniel
• Beteiligte Personen und Organisationen: Schneider, Rolf ; Albert-Ludwigs-Universität Freiburg. Mathematische Fakultät

Weitere Informationen

• DOI: 10.6094/UNIFR/167349
• URN: urn:nbn:de:bsz:25-freidok-1673493
• Rechteinformation: Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Letzte Aktualisierung: 25.03.2025, 13:46 MEZ