Eine Erweiterung der Croftonschen Formeln für konvexe Körper

§0. Einführung. Durch die bekannten Croftonschen Integrale können bekanntlich die Minkowskischen Quermaβintegrale konvexer Körper dargestellt werden. In der vorliegenden Note betrachten wir gewisse Erweiterungen dieser klassischen integralgeometrischen Formeln, durch die allgemeinere invariante Eikörperfunktionale gegeben sind. Es handelt sich hierbei um kinematische Integrale mit beweglichen unterdimensionalen Teilräumen, wobei passend gewahlte Funktionen ihrer Abstände vom Eikörper eingehe

Morphological Thermodynamics of Fluids: Shape Dependence of Free Energies

We examine the dependence of a thermodynamic potential of a fluid on the geometry of its container. If motion invariance, continuity, and additivity of the potential are fulfilled, only four morphometric measures are needed to describe fully the influence of an arbitrarily shaped container on the fluid. These three constraints can be understood as a more precise definition for the conventional term "extensive" and have as a consequence that the surface tension and other thermodynamic quantities contain, beside a constant term, only contributions linear in the mean and Gaussian curvature of the container and not an infinite number of curvatures as generally assumed before. We verify this numerically in the entropic system of hard spheres bounded by a curved wall.Comment: 4 pages, 3 figures, accepted for publication in PR

Eutactic quantum codes

We consider sets of quantum observables corresponding to eutactic stars. Eutactic stars are systems of vectors which are the lower dimensional ``shadow'' image, the orthogonal view, of higher dimensional orthonormal bases. Although these vector systems are not comeasurable, they represent redundant coordinate bases with remarkable properties. One application is quantum secret sharing.Comment: 6 page

Density functional theory for hard-sphere mixtures: the White-Bear version Mark II

In the spirit of the White-Bear version of fundamental measure theory we derive a new density functional for hard-sphere mixtures which is based on a recent mixture extension of the Carnahan-Starling equation of state. In addition to the capability to predict inhomogeneous density distributions very accurately, like the original White-Bear version, the new functional improves upon consistency with an exact scaled-particle theory relation in the case of the pure fluid. We examine consistency in detail within the context of morphological thermodynamics. Interestingly, for the pure fluid the degree of consistency of the new version is not only higher than for the original White-Bear version but also higher than for Rosenfeld's original fundamental measure theory.Comment: 16 pages, 3 figures; minor changes; J. Phys.: Condens. Matter, accepte

Lines pinning lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.Comment: 27 pages, 10 figure

On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure