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.     

The mixed Lp affine surface areas for functions

In this paper, we propose a definition of a general mixed L_p Affine surface area, ?n ≠ p ∈ ?, for multiple functions. Our definition is di?erent from and is “dual” to the one in [11] by Caglar and Ye. In particular, our definition makes it possible to establish an integral formula for the general mixed L_p Affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved.     

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     

Projection bodies in complex vector spaces

The space of Minkowski valuations on an m-dimensional complex vector space which are continuous, translation invariant and contravariant under the complex special linear group is explicitly described. Each valuation with these properties is shown to satisfy geometric inequalities of the Brunn–Minkowski, Aleksandrov–Fenchel and Minkowski type.     

Splitting and pinching for convex sets

We consider noncompact, closed and convex sets with nonvoid interior in Euclidean space. It is shown that if such a set has one curvature measure sufficiently close to the boundary measure, then it is congruent to a product of a vector space and a compact convex body. Related stability and characterization theorems for orthogonal disc cylinders are proved. Our arguments are based on the Steiner-Schwarz symmetrization processes and generalized Minkowski integral formulas.     

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     

Positions of convex bodies associated to extremal problems and isotropic measures

We show that there are close relations between extremal problems in dual Brunn-Minkowski theory and isotropic-type properties for some Borel measures on the sphere. The methods we use allow us to obtain similar results in the context of Firey-Brunn-Minkowski theory. We also study reverse inequalities for dual mixed volumes which are related with classical positions, such as ?-position or isotropic position.     

General affine surface areas

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for L? affine surface areas are established.     

The unit ball is an attractor of the intersection body operator

The intersection body of a ball is again a ball. So, the unit ball Bd⊂Rd is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach–Mazur distance. E. Lutwak asked if there is any other star-shaped body that satisfies this property. We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to Bd in Banach–Mazur distance converge to Bd in Banach–Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of Bd. We will also discuss a harmonic analysis version of this question, which studies the Radon transforms of powers of a given function.     

The classification problem for S-local torsion-free abelian groups of finite rank

Suppose that n?2 and that S, T are sets of primes. Then the classification problem for the S-local torsion-free abelian groups of rank n is Borel reducible to the classification problem for the T-local torsion-free abelian groups of rank n if and only if S⊆T.     

Higher integrality conditions, volumes and Ehrhart polynomials

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.     

On the Sobolev distance of convex bodies

Summary LetK ^d denote the cone of all convex bodies in the Euclidean spaceK ^d . The mappingK h _K of each bodyK K ^d onto its support function induces a metric _w onK ^d by" _w (K, L)h _L –h _{K w} where _w is the Sobolev I-norm on the unit sphere . We call _w (K, L) the Sobolev distance ofK andL. The goal of our paper is to develop some fundamental properties of the Sobolev distance.     

Duality of convex bodies

The paper presents a category theoretical approach to the notion of duality of convex bodies. Using results of I. Barany (Acta Sci. Math. (Szeged)52 (1988), 93–100), we define and study metric duality , whose advantage is that congruent convex bodies have congruent duals.Dedicated to Professor Helmut Salzmann on the occasion of his 65th birthday     

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.     

Critical metrics for quadratic functionals in the curvature on 4-dimensional manifolds

We study critical metrics for the squared L²-norm functionals of the curvature tensor, the Ricci tensor and the scalar curvature by making use of a curvature identity on 4-dimensional Riemannian manifolds.     

Enclosed convex hypersurfaces with maximal affine area

In this paper we study the regularity of closed, convex surfaces which achieve maximal affine area among all the closed, convex surfaces enclosed in a given domain in the Euclidean 3-space. We prove the C^1,α regularity for general domains and C^1,1 regularity if the domain is uniformly convex. This work is supported by the Australian Research Council. Research of Sheng was also supported by ZNSFC No. 102033. On leave from Zhejiang University.     

Sections and projections of homothetic convex bodies

For a pair of convex bodies K₁ and K₂ in Euclidean space , n ≥ 3, possibly unbounded, we show that K₁ is a translate of K₂ if either of the following conditions holds: (i) the orthogonal projections of K₁ on 2-dimensional planes are translates of the respective orthogonal projections of K₂, (ii) there are points p₁ ∈K₁ and p₂ ∈K₂ such that for every pair of parallel 2-dimensional planesL₁and L₂ through p₁ and p₂, respectively, the section K₁ ∩ L₁is a translate of K₂ ∩ L₂.