Characterizing the grain structure of polycrystalline material is an important task in material science. The present paper introduces the concept of generalized balanced power diagrams as a concise alternative to voxelated mappings. Here, each grain is represented by (measured approximations of) its center-of-mass position, its volume and, if available, by its second-order moments (in the non-equiaxed case). Such parameters may be obtained from 3D x-ray diffraction. As the exact global optimum of our model results from the solution of a suitable linear program it can be computed quite efficiently. Based on verified real-world measurements we show that from the few parameters per grain (3, respectively 6 in 2D and 4, respectively 10 in 3D) we obtain excellent representations of both equiaxed and non-equiaxed structures. Hence our approach seems to capture the physical principles governing the forming of such polycrystals in the underlying process quite well.

Abstract. The Tarim River basin, located in Xinjiang, NW China, is the largest endorheic river basin in China and one of the largest in all of Central Asia. Due to the extremely arid climate, with an annual precipitation of less than 100 mm, the water supply along the Aksu and Tarim rivers solely depends on river water. This is linked to anthropogenic activities (e.g., agriculture) and natural and semi-natural ecosystems as both compete for water. The ongoing increase in water consumption by agriculture and other human activities in this region has been enhancing the competition for water between human needs and nature. Against this background, 11 German and 6 Chinese universities and research institutes have formed the consortium SuMaRiO (Sustainable Management of River Oases along the Tarim River; http://www.sumario.de), which aims to create a holistic picture of the availability of water resources in the Tarim River basin and the impacts on anthropogenic activities and natural ecosystems caused by the water distribution within the Tarim River basin. On the basis of the results from field studies and modeling approaches as well as from suggestions by the relevant regional stakeholders, a decision support tool (DST) will be implemented that will then assist stakeholders in balancing the competition for water, Published by Copernicus Publications on behalf of the European Geosciences Union. 84 C. Rumbaur et al.: Sustainable management of river oases along the Tarim River acknowledging the major external effects of water allocation to agriculture and to natural ecosystems. This consortium was formed in 2011 and is funded by the German Federal Ministry of Education and Research. As the data collection phase was finished this year, the paper presented here brings together the results from the fields from the disciplines of climate modeling, cryology, hydrology, agricultural sciences, ecology, geoinformatics, and social sciences in order to present a comprehensive picture of the effects of different water availability schemes on anthropogenic activities and natural ecosystems along the Tarim River. The second objective is to present the project structure of the whole consortium, the current status of work (i.e., major new results and findings), explain the foundation of the decision support tool as a key product of this project, and conclude with application recommendations for the region. The discharge of the Aksu River, which is the major tributary of the Tarim, has been increasing over the past 6 decades. From 1989 to 2011, agricultural area more than doubled: cotton became the major crop and there was a shift from small-scale to large-scale intensive farming. The ongoing increase in irrigated agricultural land leads to the increased threat of salinization and soil degradation caused by increased evapotranspiration. Aside from agricultural land, the major natural and semi-natural ecosystems are riparian (Tugai) forests, shrub vegetation, reed beds, and other grassland, as well as urban and peri-urban vegetation. ...

This paper is concerned with convex bodies in n‐dimensional lp, spaces, where each body is accessible only by a weak separation or optimization oracle. It studies the asymptotic relative accuracy, as n→∞, of polynomial‐time approximation algorithms for the diameter, width, circumradius, and inradius of a body K, and also for the maximum of the norm over K.

We study weighted clustering problems in Minkowski spaces under balancing constraints with a view towards separation properties. First, we introduce the gravity polytopes and more general gravity bodies that encode all feasible clusterings and indicate how they can be utilized to develop efficient approximation algorithms for quite general, hard to compute objective functions. Then we show that their extreme points correspond to strongly feasible power diagrams, certain specific cell complexes, whose defining polyhedra contain the clusters, respectively. Further, we characterize strongly feasible centroidal power diagrams in terms of the local optima of some ellipsoidal function over the gravity polytope. The global optima can also be characterized in terms of the separation properties of the corresponding clusterings.

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