Translative integral formulae for convex bodies

Translative versions of the principal kinematic formula for quermassintegrals of convex bodies are studied. The translation integral is shown to be a sum of Crofton type integrals of mixed volumes. As corollaries new integral formulas for mixed volumes are obtained. For smooth centrally symmetric bodies the functionals occurring in the principal translative formula are expressed by measures on Grassmannians which are related to the generating measures of the bodies.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

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.     

On the mean number of normals through a point in the interior of a convex body

Recently, Kathy Hann established bounds on the average number of normals through a point in a convex bodyK, in the cases whereK is either a polytope or sufficiently smooth. In addition, an Euler-type theorem was obtained for these particular classes of convex bodies. In the present work we show that all these statements are true for an arbitrary convex bodyK. For this purpose measure geometric tools and a general approximation technique will be essential.     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

Asymptotic formulas for the diameter of sections of symmetric convex bodies

Sharpening work of the first two authors, for every proportion λ∈(0,1) we provide exact quantitative relations between global parameters of n-dimensional symmetric convex bodies and the diameter of their random ⌊λn⌋-dimensional sections. Using recent results of Gromov and Vershynin, we obtain an “asymptotic formula” for the diameter of random proportional sections.     

Contributions to max-min convex geometry. I: Segments

We give some contributions to the theory of “max-min convex geometry”, that is, convex geometry in the semimodule over the max-min semiring R_max,min=R∪{-∞,+∞}. We introduce “elementary segments” that generalize from n=2 the horizontal, vertical or oblique segments contained in the main bisector of . We show that every segment in is a concatenation of a finite number of elementary subsegments (at most 2n-1, respectively at most 2n-2, in the case of comparable, respectively, incomparable, endpoints x,y). In this first part we study “max-min segments”, and in the subsequent second part (submitted) we study “max-min semispaces” and some of their relations to “max-min convex sets”.     

The Orlicz centroid inequality for star bodies

Lutwak, Yang and Zhang established the Orlicz centroid inequality for convex bodies and conjectured that their inequality can be extended to star bodies. In this paper, we confirm this conjecture.     

A characterization for isometries and conformal mappings of pseudo-Riemannian manifolds

Generalizing theorems of Myers-Steenrod and of Hawking, we obtain characterizations for isometries and conformal mappings of pseudo-Riemannian spaces (M, g): Define a local distance function on convex normal neighbourhoods by (p, q) =g(exp _p ^–1 q, exp _p ^–1 q). Then every homeomorphismf locally preserving these functions is an isometry. If (M, g) has indefinite signature andf locally preserves distance zero, it is a conformal diffeomorphism.     

On the vertex index of convex bodies

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K⊂Rd, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by d² smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K⊂Rd one has

     

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

For a convex body K ^d we investigate three associated bodies, its intersection body IK (for 0int K), cross-section body CK, and projection body IIK, which satisfy IKCKIIK. Conversely we prove CKconst₁(d)I(K–x) for some xint K, and IIKconst₂ (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L ^d a convex body, we take n random segments in L, and consider their Minkowski average D. We prove that, for V(L) fixed, the supremum of V(D) (with also nN arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M ^d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1p<) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (nN arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 41.     

A Möbius characterization of submanifolds in real space forms with parallel mean curvature and constant scalar curvature

In this paper, we give a Möbius characterization of submanifolds in real space forms with parallel mean curvature vector fields and constant scalar curvatures, generalizing a theorem of H. Li and C.P. Wang in [LW1].Supported by NSF of Henan, P. R. China     

Der Satz von Liouville über räumliche orthogonaltreue (winkeltreue) Abbildungen für beliebige Signatur

Ohne Zusammenfassung

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Irregularities of point distributions relative to homothetic convex bodies I

The uniformity and irregularities of point distributions can be measured by various kinds of geometric objects. In this paper we prove the existence of point sets that have relatively small irregularities with respect to homothetic copies of a fixed convex body. The results give higher dimensional alternatives to a theorem of Beck.Supported by the Alexander von Humboldt Foundation.     

A vector-sum theorem in two-dimensional space

Given a finite setX of vectors from the unit ball of the max norm in the twodimensional space whose sum is zero, it is always possible to writeX = {x₁, , x_n} in such a way that the first coordinates of each partial sum lie in [–1, 1] and the second coordinates lie in [–C, C] whereC is a universal constant.     

The Willmore conjecture for immersed tori with small curvature integral

The Willmore conjecture states that any immersion of a 2-torus into euclidean space satisfies . We prove it under the condition that the L ^p -norm of the Gaussian curvature is sufficiently small. Received: 10 June 1999     

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     

Hahn—Banach and Banach—Steinhaus theorems for convex processes

Our goal in this paper is to prove versions of the Hahn—Banach and Banach—Steinhaus theorems for convex processes. For the first theorem, we prove that each real convex process corresponds to a sublinear or superlinear functional and then we extend these. For the second theorem, we previously show that there is a seminorm naturally associated to the class of lower semi-continuous convex processes.