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

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. Received: 15 July 1999     

Ein Approximationssatz für konvexe Körper

It is shown that a convex body in n-dimensional Euclidean space can be approximated by a sequence of smooth convex bodies in such a way that the principal radii of curvature converge in a certain sense. This fact is used to characterize those first surface measures of convex bodies which belong to polytopes. Furthermore it is proved that the support function of a convex body whose first surface measure has bounded density must have continuous first partial derivatives.     

The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.     

On the structure of convex sets with symmetries

Asymmetry of a compact convex body L ì Rⁿ{\mathcal L \subset {\bf R}^n} viewed from an interior point O{\mathcal O} can be measured by considering how far L{\mathcal L} is from its inscribed simplices that contain O{\mathcal O}. This leads to a measure of symmetry s(L, O){\sigma(\mathcal L, \mathcal O)}. The interior of L{\mathcal L} naturally splits into regular and singular sets, where the singular set consists of points O{\mathcal O} with largest possible s(L, O){\sigma(\mathcal L, \mathcal O)}. In general, to calculate the singular set of a compact convex body is difficult. In this paper we determine a large subset of the singular set in centrally symmetric compact convex bodies truncated by hyperplane cuts. As a function of the interior point O{\mathcal O}, s(L, .){\sigma(\mathcal L, .)} is concave on the regular set. It is natural to ask to what extent does concavity of s(L, .){\sigma(\mathcal L, .)} extend to the whole (interior) of L{\mathcal L}. It has been shown earlier that in dimension two, s(L, .){\sigma(\mathcal L, .)} is concave on L{\mathcal L}. In this paper, we show that in dimensions greater than two, for a centrally symmetric compact convex body L{\mathcal L}, s(L, .){\sigma(\mathcal L, .)} is a non-concave function provided that L{\mathcal L} has a codimension one simplicial intersection. This is the case, for example, for the n-dimensional cube, n ≥ 3. This non-concavity result relies on the fact that a centrally symmetric compact convex body has no regular points.     

The extreme bodies in the set of plane convex bodies with a given width function

Let \(\bar K\) (w) denote the class of plane convex bodies having a width functionw. Examining the length measure of the boundary of a convex body in \(\bar K\) (w), a characterization is given for the extreme (indecomposable) bodies in \(\bar K\) (w). This is a generalization of the solution previously given by the author in Israel J. Math. (1974) for the case wherew′ is absolutely continuous.     

Singular sets of convex bodies and surfaces with generalized curvatures

In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional subsets Σ ^k of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention is given to boundaries of convex bodies inR ³. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider integral functionals depending on the curvature and the area ofK and on the curvature andH ^k of Σ ^k .     

Filtration and Finite-Dimensional Characterization of Logarithmically Convex Measures

We study the classes C(, ) and C _H(, ) of logarithmically convex measures that are a natural generalization of the notion of Boltzmann measure to an infinite-dimensional case. We prove a theorem on the characterization of these classes in terms of finite-dimensional projections of measures and describe some applications to the theory of random series.     

Absolute curvature measures,II

The absolute curvature measures for sets of positive reach in R ^d introduced in [7] satisfy the following kinematic relations: Their integrated values on the intersections with (or on the tangential projections onto) uniformly moved p-planes are constant multiples of the corresponding absolute curvature measures of the primary set. In the special case of convex bodies the first result is the so-called Crofton formula. An analogue for signed curvature measures is well known in the differential geometry of smooth manifolds, but the motion of absolute curvatures used there does not lead to this property. For the special case of smooth compact hypermanifolds our absolute curvature measures agree with those introduced by Santaló [4] with other methods.In the appendix, the section formula is applied to motion invariant random sets.     

Global C2‐Estimates for Convex Solutions of Curvature Equations

We establish a C² a priori estimate for convex hypersurfaces whose principal curvatures κ=(κ₁,…, κ_n) satisfy σ_k(κ(X))=f(X,ν(X)), the Weingarten curvature equation. We also obtain such an estimate for admissible 2‐convex hypersurfaces in the case k=2. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.© 2015 Wiley Periodicals, Inc.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

Measure-Valued Valuations and Mixed Curvature Measures of Convex Bodies

We prove a polynomial expansion for measure-valued functionals which are translation covariant on the set of convex bodies. The coefficients are measures on product spaces. We then apply this construction to the curvature measures of convex bodies and obtain mixed curvature measures for bodies in general relative position. These are used to generalize an integral geometric formula for nonintersecting convex bodies. Finally, we introduce support measures relative to a quite general structuring body B and describe connections between the different types of measures.     

Further results on increasing paths in the one-skeleton of a convex body

Various problems are considered in an attempt to generalize the simplex algorithm of linear programming to a much wider class of convex bodies than the class of convex polytopes. A conjecture of D.G. Larman and C.A. Rogers is disproved by constructing a three-dimensional convex body K with an extreme point e, so that for a certain linear functional f, there are no paths in the one-skeleton of K leading from e, along which f strictly increases. Their conjectured generalization is, however, proved for the large class of three-dimensional convex bodies, all of whose extreme points are exposed.A strong generalization of the simplex algorithm is obtained for the class of all finite-dimensional convex bodies, where, for a given exposed point e of a convex body K, it is possible to find f-strictly-increasing paths in the one-skeleton of K, leading from e, for almost all linear functionals f.Research sponsored by the British Science Research Council.     

Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.     

Reduced bodies in normed planes

We say that a convex body R of a d-dimensional real normed linear space M ^d is reduced, if Δ(P) < Δ(R) for every convex body P ⊂ R different from R. The symbol Δ(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ M ^d . We establish a number of properties of reduced bodies in M ². They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M ² is larger than Δ(R) for all directions strictly between two fixed directions and equals Δ(R) for these two directions.     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

Curvatures and currents for unions of sets with positive reach

For locally finite unions of sets with positive reach in R ^d, generalized unit normal bundles are introduced in support of a certain set additive index function. Given an appropriate orientation to the normal bundle, signed curvature measures may be defined by means of associated locally rectifiable currents (with index function as multiplicity) and specially chosen differential forms. In the case of regular sets this is shown to be equivalent to well-known classical concepts via former results. The present approach leads to unified methods in proving integral-geometric relations. Some of them are stated in this paper.     

A Universal Measure

It is shown that for any ring R of sets there is a locally convex Hausdorff topological vector space M(R) and a measure χ : R → M(R) such that for any other locally convex Hausdorff topological vector space W and any measure µ : R → W, there exists a unique continuous linear map µ : M(R) → W such that . Measure is defined in such a way that M(R)′, the space of all continuous linear functionals on M(R), corresponds to all finite, signed measures in the usual sense when R is a σ-ring. The problem of extending measures from R to the σ-ring generated by R is formulted in this setting. Properties of the universal measure χ are described. The topology of M(R) is studied via the pairing <M(R), M(R)′>. For example, the weak compact subsets of M(R)′ are characterized. This characterization gives rise to a simple proof of the Orlicz-Pettis theorem.     

The Illumination Body of Almost Polygonal Bodies

For a convex body K in R ² with illumination body K we consider an expression connected with the affine surface area of K, namely the volume differences vol₂(K ) - vol₂(K). We investigate what kind of functions can occur as such volume differences and obtain a result similar to the one obtained in the case of floating bodies.     

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n–2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.In dimensions n=2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n–2)-mean curvature integral of its boundary.In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.