An optimal extension of Marstrand’s plane-packing theorem

We prove that if F is a subset of the 2-dimensional unit sphere in $\mathbb{R}^3$, with Hausdorff dimension strictly greater than 1, and E is a subset of $\mathbb{R}^3$ such that for each $e \in F$, E contains a plane perpendicular to the vector e, then E must have positive 3-dimensional Lebesgue measure.Received: 16 April 2002     

Conical upper density theorems and porosity of measures

We study how measures with finite lower density are distributed around (n−m)-planes in small balls in Rn. We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.     

On Bandt's tangential distribution for self-similar measures

It is shown that the local geometry of a self-similar measure as captured by Bandt's average tangential distribution is the same at -almost all points of the underlying space. Moreover, for a self-similar measure explicit formulas for Bandt's tangential distribution as well as for the average density of Bedford and Fisher are derived.     

A representation theorem for maxitive measures

A maxitive measure is a nonnegative function η on a σ-algebra Σ and such that η(U_j A_j ) = sup_j η(A_j) for all countable disjoint families of sets (A_j) in Σ. A representation theorem for such measures is established, and next applied to represent Köthe function M-spaces as L_∞-spaces.     

A theorem on measures in dimension 2 and applications to vortex sheets

We find conditions under which measures belong to H⁻¹(R²)$H^{-1}({\mathbb{R}}^2)$. Next we show that measures generated by the Prandtl, Kaden as well as Pullin spirals, objects considered by physicists as incompressible flows generating vorticity, satisfy assumptions of our theorem, thus they are (locally) elements of H⁻¹(R²)$H^{-1}({\mathbb{R}}^2)$. Moreover, as a by-product, we prove an embedding of the space of Morrey type measures in H⁻¹$H^{-1}$.     

A unifying Radon-Nikodym theorem for vector measures

It is a known fact that certain derivation bases from martingales with a directed index set. On the other hand it is also true that the strong convergence of certain abstract martingales is a consequence of the Radon-Nikodym theory for vector measures (cf. Uhl, J. J., Jr., Trans. Amer. Math. Soc.145 1969, 271–285). Many other connections and applications of the latter theory with multidimensional problems in stochastic processes and representation theory are known (cf. Dinculeanu, N., Studia Math.25 1965, 181–205; Dinculeanu, N., and Foias, C., Canad. J. Math.13 1961, 529–556; Rao, M. M., Ann. Mat. pura et applicata76 1967, 107–132; Rybakov, V. I., Izv. Vys?. U?ebn. Zaved. Matematika19 1968, 92–101; Rybakov, V. I., Dokl. Akad. Nauk SSSR180 1968, 620–623). Starting from various vantage points, many authors have proposed several hypotheses for establishing abstract Radon-Nikodym theorems. In view of the great interest and importance of this problem in the areas mentioned above, it is natural to obtain a unifying result with a general enough hypothesis to deduce the various forms of the Radon-Nikodym theorem for vector measures. This should illuminate the Radon-Nikodym theory for vector measures and stimulate further work in abstract martingale problems. In this paper the first problem is attacked, leaving the martingale part and other applications for another study.The main result (Theorem 7 of Section 2) provides the desired unification and from if the Dunford-Pettis theorem, the Phillips theorem and several others are obtained. As martingale-type arguments are constantly present, a careful reader may note the easy translation of the hypothesis to the martingale convergence problem but we treat only the Radon-Nikodym problem using the language of measure theory and linear analysis.     

Variational aspects of Poncelet's theorem

We prove results about the maximum or minimum of the length of a convex polygon inscribed in an ellipse or circumscribed around it, respectively. Combining these, we obtain a new proof of Poncelet's theorem on homofocal ellipses and convex polygons.     

A fixed point theorem for moment matrices of self-similar measures

We consider the self-similar measure on the complex plane C$\mathbb{C}$ associated to an iterated function system (IFS) with probabilities. From this IFS we define an operator in a complete metric space of infinite matrices. Using the expression obtained in a previous work of the authors, we prove that this operator has as fixed point the moment matrix of the self-similar measure. As a consequence, we obtain a very efficient algorithm to compute the moment matrix of the self-similar measure. Finally, in order to estimate the rate of convergence of the algorithm, we find an upper bound of the norm of this contractive operator.     

A decomposition theorem for maxitive measures

A maxitive measure is the analogue of a finitely additive measure or charge, in which the usual addition is replaced by the supremum operation. In contrast to charges, maxitive measures often have a density. We show that maxitive measures can be decomposed as the supremum of a maxitive measure with density, and a residual maxitive measure that is null on compact sets under specific conditions.     

A Grothendieck-type theorem for the space of totally measurable functions

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X^* stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X)^* and B(Σ, X)^** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X^*) denote the Banach space of all countably additive vector measures ν: Σ → X^* of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X^*), B(Σ, X))-sequential compactness in bvca(Σ, X^*) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ(B(Σ, X)^*, B(Σ, X))-sequentially compact subset of B(Σ, X)_c^~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ(B(Σ, X)^*, B(Σ, X)^**)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ(B(Σ, X)^*, B(Σ, X))-convergent sequences in B(Σ, X)_c^~ are σ(B(Σ, X)^*, B(Σ, X)^**)-convergent.     

A general duality theorem for marginal problems

Summary Given probability spaces (X _i ,A _i ,P _i ),i=1, 2 letM(P ₁,P ₂) denote the set of all probabilities on the product space with marginalsP ₁ andP ₂ and leth be a measurable function on (X ₁×X ₂,A ₁ A ₂). In order to determine supfh dP where the supremum is taken overP inM(P ₁,P ₂), a general duality theorem is proved. Only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions. An example without any further topological or tightness assumptions. An example is given to show that the assumption of perfectness is essential. Applications to probabilities with given marginals and given supports, stochastic order and probability metrics are included.     

A differentiable structure for metric measure spaces

The main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure. This differentiable structure gives rise to a finite-dimensional L^∞ cotangent bundle over the given metric measure space and then to a Sobolev space H^1,p over the given metric measure space, the latter which is reflexive for p>1. This extends results of Cheeger (Geom. Funct. Anal. 9 (1999) (3) 428) to a wider collection of metric measure spaces.     

Tangent measure distributions of fractal measures

Tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in Euclidean spaces. The idea is, loosely speaking, to attach to every point of the set a family of random measures, called the -dimensional tangent measure distributions at the point, which describe asymptotically the -dimensional scenery seen by an observer zooming down towards this point. This tool has been used by Bandt [BA] and Graf [G] to study the regularity of the local geometry of self similar sets, but in this paper we show that its scope goes much beyond this situation and, in fact, it may be used to describe a strong regularity property possessed by every measure: We show that, for every measure on a Euclidean space and any dimension , at -almost every point, all -dimensional tangent measure distributions are Palm measures. This means that the local geometry of every dimension of general measures can be described – like the local geometry of self similar sets – by means of a family of statistically self similar random measures. We believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities. Received: 27 November 1996 / Revised version: 27 February 1998     

The doubling property of conformal measures of infinite iterated function systems

We provide sufficient conditions for the conformal measures induced by regular conformal infinite iterated function systems to satisfy the doubling property. We apply these conditions to iterated function systems derived from the continued fraction algorithm—continued fractions with restricted entries. For these systems our conditions are expressed in terms of the asymptotic density properties of the allowed entries. As examples, we give some relatively large classes of sets of continued fractions with restricted entries for which the corresponding conformal measures have the doubling property. Similarly, we give some other classes for which the conformal measure does not have the doubling property.     

A remark on a theorem of Losert

Subject to the continuum hypothesis, the Haar measure on a compact group of topological weight ₂ admits a Baire strong lifting.     

Doubling measures,monotonicity, and quasiconformality

We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure. L.V.K. was supported by an NSF Young Investigator award under grant DMS 0601926. J.-M.W. was supported by the NSF grant DMS 0400810.     

The “hot spots” conjecture for the Sierpinski gasket

There are many works on the “hot spots” conjecture for domains in Euclidean space since the conjecture was posed by J. Rauch in 1974. In this paper, using spectral decimation, we prove that the conjecture holds on the Sierpinski gasket, i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian (introduced by J. Kigami) attains its maximum and minimum on the boundary.     

Lattice measures and associated outer measures

In this paper, we consider lattice measures and introduce certain associated outer measures (not the usual induced outer measures), study their properties, and investigate the associated classes of measureable sets. We utilize some of these outer measures to characterize normality and investigate lattice separation properties; also, to extend the notion of regularity of measures to weak regularity of measures. We give applications of our results to specific topological lattices.     

A lower bound theorem for polytope pairs

V. Klee has generalized the lower bound theorem of D. Barnette for polytope pairs. He has extensively studied polytope pairs. Here, two theorems for polytope pairs, which are analogous to those of Barnette and lead to more open problems for polytope pairs, are proved. Two theorems of Klee and two of Barnette are immediate corollaries of the theorems.     

The upper and lower quantization coefficient for Markov‐type measures

We study the asymptotic quantization error for Markov‐type measures μ on a class of ratio‐specified graph directed fractals E . Assuming a separation condition for E , we show that the quantization dimension for μ of order r exists and determine its exact value in terms of spectral radius of a related matrix. We prove that the ‐dimensional lower quantization coefficient for μ is always positive. Moreover, we establish a necessary and sufficient condition for the ‐dimensional upper quantization coefficient for μ to be finite.