From <Emphasis Type="Italic">r</Emphasis>-dual sets to uniform contractions

Let \(\mathbb {M}^d\) denote the d-dimensional Euclidean, hyperbolic, or spherical space. The r-dual set of a given set in \(\mathbb {M}^d\) is the intersection of balls of radii r centered at the points of the a given set. In this paper we prove that for any set of given volume in \(\mathbb {M}^d\) the volume of the r-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser–Poulsen Conjecture states that if the centers of a family of N congruent balls in Euclidean d-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers, that is, when the pairwise distances of the two sets are separated by some positive real number. We prove a special case of the Kneser–Poulsen conjecture namely, we prove the conjecture for uniform contractions (with sufficiently large N) in \(\mathbb {M}^d\).     

On the Kneser–Poulsen conjecture in elliptic space

The Kneser–Poulsen conjecture claims that if some balls of Euclidean space are rearranged in such a way that the distances between their centers do not increase, then neither does the volume of the union of the balls. A special case of the conjecture, when the balls move continuously in such a way that the distances between the centers (weakly) decrease during the motion, is known to hold not only in Euclidean, but also in spherical and hyperbolic spaces. In the present paper, we show that this theorem cannot be extended to elliptic space by constructing three smoothly moving congruent balls with centers getting closer to one another in such a way that the volume of the union of the balls strictly increase during the motion. In spite of this counterexample, it is true that n + 1 balls in n-dimensional elliptic space cover maximal volume if the distances between the centers are all equal to the diameter π/2 of the space. The second part of the paper is devoted to the proof of this fact.

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The authors were supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T047102 and T037752.     

Strict Kneser–Poulsen conjecture for large radii

In this paper we prove the Kneser–Poulsen conjecture for the case of large radii. Namely, if a finite number of points in Euclidean space ${\mathbb{E}^n}$ is rearranged so that the distance between each pair of points does not decrease, then there exists a positive number r ₀ that depends on the rearrangement of the points, such that if we consider n-dimensional balls of radius r > r ₀ with centers at these points, then the volume of the union (intersection) of the balls before the rearrangement is not less (not greater) than the volume of the union (intersection) after the rearrangement. Moreover, the inequality is strict whenever the new point set is not congruent to the original one. Also under the same conditions we prove a similar result about surface volumes instead of volumes. In order to prove the above mentioned results we use ideas from tensegrity theory to strengthen the theorem of Sudakov (Dokl. Akad. Nauk SSSR 197:43–45, 1971), Alexander (Trans. Am. Math. Soc., 288(2):661–678, 1985) and Capoyleas and Pach (Discrete and computational geometry. American Mathematical Society, Providence, 1991), which says that the mean width of the convex hull of a finite number of points does not decrease after an expansive rearrangement of those points. In this paper we show that the mean width increases strictly, unless the expansive rearrangement was a congruence. We also show that if the configuration of centers of the balls is fixed and the volume of the intersection of the balls is considered as a function of the radius r, then the second highest term in the asymptotic expansion of this function is equal to ${-M_nr^{n-1}}$ , where M _n is the mean width of the convex hall of the centers. This theorem was conjectured by Balázs Csikós in 2009.     

The Kneser–Poulsen Conjecture for Spherical Polytopes

If a finite set of balls of radius /2 (hemispheres) in the unit sphere Sⁿ is rearranged so that the distance between each pair of centers does not decrease, then the (spherical) volume of the intersection does not increase, and the (spherical) volume of the union does not decrease. This result is a spherical analog to a conjecture by Kneser (1954) and Poulsen (1955) in the case when the radii are all equal to /2.     

Volumetric bounds for intersections of congruent balls in Euclidean spaces

We investigate the intersections of balls of radius r, called r-ball bodies, in Euclidean d-space. An r-lense (resp., r-spindle) is the intersection of two balls of radius r (resp., balls of radius r containing a given pair of points). We prove that among r-ball bodies of a given volume, the r-lense (resp., r-spindle) has the smallest inradius (resp., largest circumradius). In general, we upper (resp., lower) bound the intrinsic volumes of r-ball bodies of a given inradius (resp., circumradius). This complements and extends some earlier results on volumetric estimates for r-ball bodies.

     

Contact graphs of unit sphere packings revisited

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have. In this paper, improving earlier estimates, we prove that the number of touching pairs in an arbitrary packing of n unit balls in ${\mathbb{E}^{3}}$ is always less than ${6n - 0.926n^{\frac{2}{3}}}$ . Moreover, as a natural extension of the above problem, we propose to study the maximum number of touching triplets (resp., quadruples) in an arbitrary packing of n unit balls in Euclidean 3-space. In particular, we prove that the number of touching triplets (resp., quadruples) in an arbitrary packing of n unit balls in ${\mathbb{E}^3}$ is at most ${\frac{25}{3}n}$ (resp., ${\frac{11}{4}n}$ ).     

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.     

无穷维欧氏空间中球体及球面的Lebesgue测度

本分别给出了使在无穷维欧氏空间中球体和球面具有有限的,但又不是无穷小的测度的半径集合。     

An interior points algorithm for the convex feasibility problem

The problem of finding a point in the intersection of a finite family of convex sets in the Euclidean space R″ is considered here. We present a general algorithmic scheme which employs projections onto separating hyperplanes instead of projections onto the convex sets. This scheme includes the method of successive projections of Gubin et al., USSR Comp. Math. and Math. Phys. 7 (1967), 1–24, as a special case. A different realization proposed here is capable of handling the problem when the sets are solid and an interior point of each set is available. This alternative algorithm may, in certain cases, be more attractive than the method of Gubin et al.     

Yau's uniformization conjecture for manifolds with non-maximal volume growth

The well-known Yau's uniformization conjecture states that any complete noncompact K¨ahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress.In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C_1~n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that,under bounded curvature conditions, C_1~n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on K¨ahler manifolds with minimal volume growth.     

Bregman distances and Klee sets

In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version is a famous open problem. In this paper, we consider Klee sets from a new perspective. Rather than measuring distance induced by a norm, we focus on the case when distance is meant in the sense of Bregman, i.e., induced by a convex function. When the convex function has sufficiently nice properties, then–analogously to the Euclidean distance case–every Klee set must be a singleton. We provide two proofs of this result, based on Monotone Operator Theory and on Nonsmooth Analysis. The latter approach leads to results that complement the work by Hiriart-Urruty on the Euclidean case.     

Functorial methods in the theory of group representations I

We introduce a candidate for the group algebra of a Hausdorff group which plays the same role as the group algebra of a finite group. It allows to define a natural bijection betweenk-continuous representations of the group in a Hilbert space and continuous representations of the group algebra. Such bijections are known, but to our knowledge only for locally compact groups. We can establish such a bijection for more general groups, namely Hausdorff groups, because we replace integration techniques by functorial methods, i.e., by using a duality functor which lives in certain categories of topological Banach balls (resp., unit balls of Saks spaces).This paper was written while the authors were supported by the Swiss National Science Foundation under grant 21-33644.92.     

Anwendungen der De Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie

Various questions of classical differential geometry lead to the problem of determining all those hypersurfaces of euclidean space for which the covariant derivative of the second fundamental form vanishes identically. In the first part of this note, we investigate these hypersurfaces; the result is summarized in the theorem of the opening paragraph, whose proof is based in the decomposition theorem of De Rham (see e.g. [6], pp. 180–186). Applying this result in the second part of the note, we give local characterizations of the Euclidean sphere.     

Minimal shells containing a convex surface in Minkowski space

Summary LetK be the unit ball of a Minkowski space (finite dimensional Banach space). AK-shell is the closed set of all points between two concentric balls of the space. We consider different assignments of size to aK-shell and investigate theK-shells with minimum size which contain a given convex surface. Our results extend to Minkowski geometry classical results on minimal shells in Euclidean space. This article was processed by the author using the LATEX style file from Springer-Verlag.     

A Formulation of the Kepler Conjecture

This paper is the second in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. The top level structure of the proof is described. A compact topological space is described. Each point of this space can be described as a finite cluster of balls with additional combinatorial markings. A continuous function on this compact space is defined. It is proved that the Kepler conjecture will follow if the value of this function is never greater than a given explicit constant.     

The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

Symmetry,oriented matroids and two conjectures of Michel Las Vergnas

The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with Euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zonotopes. Both sections involve arguments concerning simplicial regions illustrating, once more, the fundamental importance of the simplex conjecture of Las Vergnas.     

Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups

We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.     

Large sieve inequalities for -forms in the conductor aspect

Duke and Kowalski in [A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)] derive a large sieve inequality for automorphic forms on GL(n) via the Rankin–Selberg method. We give here a partial complement to this result: using some explicit geometry of fundamental regions, we prove a large sieve inequality yielding sharp results in a region distinct to that in [Duke and Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)]. As an application, we give a generalization to GL(n) of Duke's multiplicity theorem from [Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices (2) (1995) 99–109 (electronic)]; we also establish basic estimates on Fourier coefficients of GL(n) forms by computing the ramified factors for GL(n)×GL(n) Rankin–Selberg integrals.