Local tensor valuations

The local Minkowski tensors are valuations on the space of convex bodies in Euclidean space with values in a space of tensor measures. They generalize at the same time the intrinsic volumes, the curvature measures and the isometry covariant Minkowski tensors that were introduced by McMullen and characterized by Alesker. In analogy to the characterization theorems of Hadwiger and Alesker, we give here a complete classification of all locally defined tensor measures on convex bodies that share with the local Minkowski tensors the basic geometric properties of isometry covariance and weak continuity

H\"older continuity for support measures of convex bodies

The support measures of a convex body are a common generalization of the curvature measures and the area measures. With respect to the Hausdorff metric on the space of convex bodies, they are weakly continuous. We provide a quantitative improvement of this result, by establishing a H\"older estimate for the support measures in terms of the bounded Lipschitz metric, which metrizes the weak convergence. Specializing the result to area measures yields a reverse counterpart to earlier stability estimates, concerning Minkowski's existence theorem for convex bodies with given area measure.Comment: The manuscript is an extended and improved version of the second part of the manuscript number arxiv:1310.151

Rotation covariant local tensor valuations on convex bodies

For valuations on convex bodies in Euclidean spaces, there is by now a long series of characterization and classification theorems. The classical template is Hadwiger's theorem, saying that every rigid motion invariant, continuous, real-valued valuation on convex bodies in $\mathbb{R}^n$ is a linear combination of the intrinsic volumes. For tensor-valued valuations, under the assumptions of isometry covariance and continuity, there is a similar classification theorem, due to Alesker. Also for the local extensions of the intrinsic volumes, the support, curvature and area measures, there are analogous characterization results, with continuity replaced by weak continuity, and involving an additional assumption of local determination. The present authors have recently obtained a corresponding characterization result for local tensor valuations, or tensor-valued support measures (generalized curvature measures), of convex bodies in $\mathbb{R}^n$. The covariance assumed there was with respect to the group ${\rm O}(n)$ of orthogonal transformations. This was suggested by Alesker's observation, according to which in dimensions $n> 2$, the weaker assumption of ${\rm SO}(n)$ covariance does not yield more tensor valuations. However, for tensor-valued support measures, the distinction between proper and improper rotations does make a difference. The present paper considers, therefore, the local tensor valuations sharing the previously assumed properties, but with ${\rm O}(n)$ covariance replaced by ${\rm SO}(n)$ covariance, and provides a complete classification. New tensor valued support measures appear only in dimensions two and three

Large faces in Poisson hyperplane mosaics

A generalized version of a well-known problem of D. G. Kendall states that the zero cell of a stationary Poisson hyperplane tessellation in ${\mathbb{R}}^d$, under the condition that it has large volume, approximates with high probability a certain definite shape, which is determined by the directional distribution of the underlying hyperplane process. This result is extended here to typical $k$-faces of the tessellation, for $k\in\{2,...,d-1\}$. This requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction.Comment: Published in at http://dx.doi.org/10.1214/09-AOP510 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Assessing proposals for a transatlantic free trade area.

Initiatives to reduce transatlantic trade barriers or to harmonize trade-related domestic policies in the EU and the US appear regularly on the agenda of policy makers. The last decade saw also considerable steps in transatlantic economic cooperation focusing on special aspects. In February 2002, a new call for a study on the benefits of a transatlantic free trade area (TAFTA) was made by the President of the EU Council to facilitate further liberalization schemes. This article examines recent developments in transatlantic economic policies, discusses changes in approaches in transatlantic regionalism and presents estimates of the economic consequences of transatlantic liberalization. Given the expected small benefits of a TAFTA and the induced costs for multilateral liberalization negotiations, the article discusses alternatives to TAFTA and argues for a multilateral approach, eventually being accompanied by some sort of open regionalism.Außenhandelsliberalisierung; Freihandelszone; EU-Staaten; USA;

Marketing Wine on the Web

E-commerce is penetrating agriculture, particularly for selling products directly to consumers. The wine industry is a case in point. The industry has long-term experience in direct marketing. Many wineries welcome patrons at their premises for wine tasting and for selling wine to them. Moreover, where the practice is allowed, shipping wine directly to consumers without the assistance of trade intermediaries is a significant sales channel. E-commerce was adopted early in the wine industries of wired high-income countries and the wine industry provides an opportunity for studying the adoption, use, and impact of e-commerce. Moreover, because e-commerce has not spread evenly through all branches of agriculture, lessons learned from the wine industry may provide useful insights for entrepreneurs and policy makers concerned with sections of agriculture or the food industry where e-commerce adoption lags behind. The specific objectives of the dissertation research project which we report here therefore were: (1) to assess the extent of e-commerce diffusion in the wine industries of Australia, California, Germany; (2) to identify e-commerce practices used by wineries for marketing wine; (3) to explain differences in wineries' e-commerce practices, and (4) to derive insights and implications for sections of agriculture that lag behind in ecommerce adoption.Marketing,

On the Complexity of Reconstructing Chemical Reaction Networks

The analysis of the structure of chemical reaction networks is crucial for a better understanding of chemical processes. Such networks are well described as hypergraphs. However, due to the available methods, analyses regarding network properties are typically made on standard graphs derived from the full hypergraph description, e.g.\ on the so-called species and reaction graphs. However, a reconstruction of the underlying hypergraph from these graphs is not necessarily unique. In this paper, we address the problem of reconstructing a hypergraph from its species and reaction graph and show NP-completeness of the problem in its Boolean formulation. Furthermore we study the problem empirically on random and real world instances in order to investigate its computational limits in practice