On the Lp Minkowski Problem for Polytopes

Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data L_p Minkowski problem for all p > 1.     

Arsenic content of hair and bones in the case of the suspected poisoning of R. Brun,first mayor of the city of Zurich

On the occasion of the 1970 opening of the grave of R. Brun, which is suspected to have been poisoned in 1360, samples of hair, bones and soil were taken and analysed for arsenic by neutron activation. Whereas the arsenic content of the hair was normal (0.2 to 0.3 ppm), the bones gave results of approximately 2 ppm, which is far above the value expected for healthy people (approximately 0.1 ppm). However, bones from the same grave (but not belonging to the suspected victim showed similar high values. These high values are thought to be due to an arsenic uptake from the soil and the evidence for an arsenic poisoning is under these circumstances rejected.     

Hydrogen-deuterium exchange experiments to probe the decomposition reaction of sodium alanate

NaAlH(4) is the archetypical complex hydride for hydrogen storage. The extraordinary effect of dopants on the sorption kinetics triggered the investigation of this empirical finding. In this paper, a short review of the state of the art is given. To gain further understanding of the mechanisms involved we label the interacting species during the sorption process. This was experimentally realized by hydrogen-deuterium exchange measurements during the decomposition of NaAlH(4) followed by thermogravimetry, Raman spectroscopy and mass spectrometry. By these experiments we are able to obtain specific information on the diffusing species and formation of intermediates. The activation energy of tracer diffusion in NaAlH(4) is found to be 0.28 eV. The results are evidence for a vacancy-mediated desorption process of NaAlH(4).     

Faces of Poisson�CVoronoi mosaics

For a stationary Poisson?CVoronoi tessellation in Euclidean d-space and for ${k\in \{1,\dots,d\}}$ , we consider the typical k-dimensional face with respect to a natural centre function. We express the distribution of this typical k-face in terms of a certain Poisson process of closed halfspaces in a k-dimensional space. Then we show that, under the condition of large inradius, the relative boundary of the typical k-face lies, with high probability, in a narrow spherical annulus.     

First‐order Raman scattering of the MAX phases Ta4AlC3, Nb4AlC3, Ti4AlN3, and Ta2AlC

Herein, we report on the Raman spectra of the following ternary hexagonal carbides and nitrides (MAX phases): Ta₄AlC₃, Ta₂AlC and Ti₄AlN₃. We also present the Raman‐active modes of α‐ and β‐Ta₄AlC₃, Nb₄AlC₃ and Ti₄AlN₃, – also referred to as the 413 MAX phases – as predicted from first principles calculations using density functional theory. We compare the obtained experimental and calculated results with previous studies on Ta₂AlC and Ti₄AlN₃. The vibrational behavior associated with the Raman‐active modes for the 413 phases has been identified for the first time. In general, the agreement is good between theory and experiment. The experimental and calculated results indicate that the modes at low wavenumbers ‐ dominated by the Al atoms ‐ are a weak function of chemistry and the differences in energy can be traced to variations in the reduced mass. The modes at higher wavenumbers are dominated by the C and N atoms and show a strong dependence on the unit cell chemistry, with the Ta–C bond being stiffer than the Nb–C bond, which is in turn stiffer than Ti–N. Copyright © 2011 John Wiley & Sons, Ltd.     

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes.     

Hessian measures of semi-convex functions and applications to support measures of convex bodies

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of σ-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies (sets of positive reach) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions. Among these are a Crofton formula, results on the absolute continuity of Hessian measures, and a duality theorem which relates the Hessian measures of a convex function to those of the conjugate function. In particular, it turns out that curvature and surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K. Received: 15 July 1999     

Lp空间Kantorovich-Vertesi算子逼近的Jackson型估计

考虑了Kantorovich-Vertesi有理插值型算子L^*n,s（f,X,x）对L^p[-1,1]（1≤p≤∞）空间函数逼近的Jackson型估计。并获得了如下逼近阶：‖L^*n,s(f,X,x)-f(x)‖L^p[-1,1]≤Cp,sw(f,1/n 2)L^p[-1,1] （s＞2）。     

Intersections and Translative Integral Formulas for Boundaries of Convex Bodies

Let K, L ⊂ ℝⁿ be two convex bodies with non–empty interiors and with boundaries ∂K, ∂L, and let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t ↦ χ(∂K ∩ (∂L + t)) and t ↦ χ(∂K ∩ (L + t)), t ∈ ℝⁿ, with respect to an n–dimensional Haar measure of ℝⁿ can be expressed in terms of certain mixed volumes of K and L. In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let K_r, L_r denote the outer parallel bodies of K, L at distance r ≥ 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of ∂K_r ∩ ∂L_r and ∂K_r ∩ L_r, respectively, is independent of r ≥ 0 if K° ∩ L° ≠ ∅︁ and if ∂K and ∂L intersect almost transversally.As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey .