Absolute continuity for curvature measures of convex sets II

Hug, Daniel

doi:10.1007/pl00004765

Cited by 14 publications

(11 citation statements)

References 48 publications

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“…The formula was proved, in a different but equivalent formulation, by Petkantschin [15, formula (48)]; it appears also in [18, formula (14.40)]. The case r = n − 1 is Lemma 5.6 in [8], with a different proof. The general case can also be obtained by a straightforward extension of the argument used there.…”

Section: Integration Over Grassmanniansmentioning

confidence: 99%

Integral geometry of tensor valuations

Hug

Schneider

Schuster³

2008

Advances in Applied Mathematics

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We prove a complete set of integral geometric formulas of Crofton type (involving integrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Minkowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger's general integral geometric theorem, the Crofton formulas yield also kinematic formulas for Minkowski tensors. The explicit calculations of integrals over affine Grassmannians require several integral geometric and combinatorial identities. The latter are derived with the help of Zeilberger's algorithm.

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Section: Integration Over Grassmanniansmentioning

confidence: 99%

Integral geometry of tensor valuations

Hug

Schneider

Schuster³

2008

Advances in Applied Mathematics

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“…Recall that the support function h K of the convex body K is twice differentiable for almost every u ∈ S n−1 . It is known (see [12], [13], [14] for additional information, references and related results concerning area measures and their densities) that f…”

Section: Preliminaries and Notationmentioning

confidence: 99%

Functions with isotropic sections

Purnaras¹,

Saroglou²

2019

Preprint

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We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if n ≥ 3, g : S n−1 → R is an even bounded measurable function, U is an open subset of S n−1 and the restriction (section) of f onto any great sphere perpendicular to U is isotropic, then C(g)| U = c + a, • and R(g)| U = c ′ , for some fixed constants c, c ′ ∈ R and for some fixed vector a ∈ R n . Here, C(g) denotes the cosine transform and R(g) denotes the Funk transform of g. However, we show that g does not need to be equal to a constant almost everywhere in U ⊥ := u∈U (S n−1 ∩u ⊥ ). For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest.

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“…The investigation of the connection between measure theoretic properties of these measures and geometric properties of the underlying sets represents a major theme in this field; see [40] and the survey given in Section 1 of [29]. Another main topic is the study of integral geometric formulae such as principal kinematic formulae or Crofton formulae and their applications to stochastic geometry.…”

Section: Introductionmentioning

confidence: 99%

Hessian measures of semi-convex functions and applications to support measures of convex bodies

Colesanti¹,

Hug²

2000

manuscripta math.

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This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of σ-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies (sets of positive reach) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions. Among these are a Crofton formula, results on the absolute continuity of Hessian measures, and a duality theorem which relates the Hessian measures of a convex function to those of the conjugate function. In particular, it turns out that curvature and surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K. IntroductionThe theory of curvature and surface area measures plays a central rôle in the classical theory of convex bodies. The importance of these measures is due to the fact that they relate classical convexity to integral and stochastic geometry, for which they are indispensible tools. But they are also relevant for such diverse subjects as the theory of valuations on convex bodies [38]

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Absolute continuity for curvature measures of convex sets II

Cited by 14 publications

References 48 publications

Integral geometry of tensor valuations

Integral geometry of tensor valuations

Functions with isotropic sections

Hessian measures of semi-convex functions and applications to support measures of convex bodies

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