The Brunn–Minkowski Inequality for the <Emphasis Type="Italic">n</Emphasis>-dimensional Logarithmic Capacity of Convex Bodies

We define the n-dimensional logarithmic capacity for convex bodies in Rⁿ, with n2; then, for this quantity, we prove a Brunn–Minkowski type inequality, and we characterize the corresponding equality case. Mathematics Subject Classifications (2000) 31C15, 31A35, 52A20, 39B62.     

A sharp isoperimetric inequality for rank one symmetric spaces

 We prove an isoperimetric inequality for compact, regular domains in rank one symmetric spaces, which is sharp for geodesic balls. Besides volume and area of a given domain, some weak information about the second fundamental form of its boundary is involved. Received: 2 September 2002 / Revised version: 10 December 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 53C35, 52A40, 51M25     

An Application of Logic to Combinatorial Geometry: How Many Tetrahedra are Equidecomposable with a Cube?

It is known (see Rapp [9]) that elementary geometry with the additional quantifier “there exist uncountably many” is decidable. We show that this decidability helps in solving the following problem from combinatorial geometry: does there exist an uncountable family of pairwise non-congruent tetrahedra that are n-equidecomposable with a cube? Mathematics Subject Classification: 03B25, 03C80, 51M04, 52B05, 52B10.     

Geometry of Log-concave Functions and Measures

We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn–Minkowski and the Blaschke–Santaló inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman’s quotient of subspace theorem, and present a functional version of the Urysohn inequality.Mathematics Subject Classiffications (2000). 52A20, 52A40, 46B07     

Edge Matrix of Hyperbolic Simplices

In this paper, by using the dual problem which was solved by Feng Luo (Geom. Dedicata 64 (1997), 277–282) and a new method, we give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a hyperbolic n-simplex. By using determinants, we also give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a spherical n-simplex.Mathematics Subject Classifications (2000). 51M04, 51M05, 51M20, 51M25, 52A38, 52A37, 52B10.     

The divergence theorem and the Laplacian in Minkowski space

Let (X, B) be a Minkowski space (finite-dimensional Banach space) with unit ball B. Using a Minkowski definition of unit normal to a hypersurface, a Minkowski analogue of Euclidean divergence is defined. We show that the divergence theorem holds. Using the Minkowski divergence, a Minkowski Laplacian is defined. We prove that this Laplacian is a second-order, constant-coefficient, elliptic, differential operator. Furthermore, the symbol of this Laplacian is computed and used to associate a natural Euclidean structure with (X, B).Supported, in part, by NSERC Operating Grant #4066.     

Inversion of adjunction for non-degenerate hypersurfaces

 We prove a precise inversion of adjunction formula for the log variety (ℂ ^{d +1},X), where X is a non-degenerate hypersurface. As a corollary, the minimal log discrepancies of non-degenerate normal hypersurface singularities are bounded by dimension. Received: 17 September 2002 / Revised version: 22 November 2002 Published online: 14 February 2003 Current address: DPMMS, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England. e-mail: f.ambro@dpmms.cam.ac.uk Mathematics Subject Classification (2000): Primary 14B05; Secondary 14M25, 52B20     

Finite element convergence for singular data

Convergence of the finite element solutionu ^h of the Dirichlet problem u= is proved, where is the Dirac -function (unit impulse). In two dimensions, the Green's function (fundamental solution)u lies outsideH ¹, but we are able to prove that . Since the singularity ofu is logarithmic, we conclude that in two dimensions the function log can be approximated inL ² near the origin by piecewise linear functions with an errorO (h). We also consider the Dirichlet problem u=f, wheref is piecewise smooth but discontinuous along some curve. In this case,u just fails to be inH ^5/2, but as with the approximation to the Green's function, we prove the full rate of convergence:u–u ^h ₁=O (h ^8/2) with, say, piecewise quadratics.     

On the Normality of Cayley Digraphs of Valency 2 on Non-Abelian Group of Odd Order

In this paper, we prove that the Cayley digraph = Cay(G, S) of valency 2 on non-abelian group G of odd order is normal if the automorphism group of A(), a graph constructed from by using the method presented in the paper, is primitive on the vertices set V(A(). We also prove that the Cayley digraphs of valency 2 on non-abelian group of order pq² are normal, where p and q are distinct odd primes.AMS Subject Classification (2000) 05C25 20B25Supported by the National Natural Science Foundation of China (Grant no. 19971086) and the Doctoral Program Foundation of the National Education Department of China.     

Spherical containment and the Minkowski dimension of partial orders

The recent work on circle orders generalizes to higher dimensional spheres. As spherical containment is equivalent to causal precedence in Minkowski space, we define the Minkowski dimension of a poset to be the dimension of the minimal Minkowski space into which the poset can be embedded; this isd if the poset can be represented by containment with spheresS ^d–2 and of no lower dimension. Comparing this dimension with the standard dimension of partial orders we prove that they are identical in dimension two but not in higher dimensions, while their irreducible configurations are the same in dimensions two and three. Moreover, we show that there are irreducible configurations for arbitrarily large Minkowski dimension, thus providing a lower bound for the Minkowski dimension of partial orders.     

The Brunn-Minkowski inequality for <Emphasis Type="Italic">p</Emphasis>-capacity of convex bodies

In this paper we prove the Brunn-Minkowski inequality for the p-capacity of convex bodies (i.e convex compact sets with non-empty interior) in R ⁿ , for every p(1,n). Moreover we prove that the equality holds in such inequality if and only if the involved bodies coincide up to a translation and a dilatation. Mathematics Subject Classification (2000):35J60, 31B15, 39B62, 52A40     

A Remarkable Measure Preserving Diffeomorphism between two Convex Bodies in ℝ n

We prove that for any two convex open bounded bodies K and T there exists a diffeomorphism f : K T preserving volume ratio (i.e. with constant determinant of the Jacobian) and such that the Minkowski sum K + T { x + f (x) | x K }. As an application of this method, we prove some of the Alexandov–Fenchel inequalities.     

Dimensions of random recursive sets

We prove a theorem that generalizes the equality among the packing, Hausdorff, and upper and lower Minkowski dimensions for a general class of random recursive constructions, and apply it to constructions with finite memory. Then we prove an upper bound on the packing dimension of certain random distribution functions on [0, 1]. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 20–26.     

Self-packing of centrally symmetric convex bodies in ℝ2

LetB be a compact convex body symmetric around0 in ℝ² which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radiusρ(m,B) is the smallestt such thatt B can be packed withm translates of the interior ofB. Form≤6 we show that the self-packing radiusρ(m,B)=1+2/α(m,B) whereα(m,B) is the Minkowski length of the side of the largest equilateralm-gon inscribed inB (measured in the Minkowski metric determined byB). We showρ(6,B)=ρ(7,B)=3 for allB, and determine most of the largest and smallest values ofρ(m,B) form≤7. For allm we have

PAC fields over finitely generated fields

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K. Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. This work constitutes a part of the Ph.D. dissertation of L. Bary-Soroker done at Tel Aviv University under the supervision of Prof. Dan Haran.     

Determination of generalized quadrangles with distinct elation points

In this paper, we classify the finite generalized quadrangles of order (s,t), s,t > 1, which have a line L of elation points, with the additional property that there is a line M not meeting L for which {L, M} is regular. This is a first fundamental step towards the classification of those generalized quadrangles having a line of elation points. Mathematics Subject Classification (2000): 51E12, 51E20, 20B25, 20E42     

The size of a Minkowski ellipse that contains the unit ball

In this paper we study the minimum radius of Minkowski ellipses (with antipodal foci on the unit sphere) necessary to contain the unit ball of a (normed or) Minkowski plane. We obtain a general upper bound depending on the modulus of convexity, and in the special case of a so-called symmetric Minkowski plane (a notion that we will recall in the paper) we prove a lower bound, and also we obtain that 3 is the exact upper bound.     

Convex Bodies, Graphs and Partial Orders

A convex corner is a compact convex down-set of full dimensionin Rⁿ. Convex corners arise in graph theory, for instance asstable set polytopes of graphs. They are also natural objects of study in geometry, as they correspond to 1-unconditionalnorms in an obvious way. In this paper, we study a parameterof convex corners, which we call the content, that is relatedto the volume. This parameter has appeared implicitly before:both in geometry, chiefly in a paper of Meyer (Israel J. Math.} 55 (1986) 317–327) effectively using content to givea proof of Saint-Raymond's Inequality on the volume product of a convex corner, and in combinatorics, especially in apaper of Sidorenko (Order} 8 (1991) 331–340) relatingcontent to the number of linear extensions of a partial order.One of our main aims is to expose connections between workin these two areas. We prove many new results, giving in particular various generalizations of Saint-Raymond's Inequality. Contentalso behaves well under the operation of pointwise product oftwo convex corners; our results enable us to give counter-examplesto two conjectures of Bollobás and Leader Oper. TheoryAdv. Appl. 77 (1995) 13–24) on pointwise products. 1991Mathematics Subject Classification: 52C07, 51M25, 52B11, 05C60,06A07.     

Centrally symmetric generators in toric Fano varieties

We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has ``many' pairs of centrally symmetric vertices. Mathematics Subject Classification (2000):14J45, 14M25, 52B20     

Eigenvalue Distribution of Elliptic Operators of Second Order with Neumann Boundary Conditions in a Snowflake Domain

Spectral properties of strongly elliptic operators of second order on bounded snowflake domains without W¹₂–extension property are investigated. We prove that the operators have a pure point spectrum and the asymptotic eigenvalue distributions for the counting function N(λ) are of Weyl type. It is shown that the remainder estimate of N(λ) for Dirichlet and Neumann boundary conditions depends on the inner and outer Minkowski dimension of the boundary ∂Ω, respectively.