Let a quadratic form on 𝔼d be represented by its coefficient vector in 𝔼(1/2)d(d+1). Then, to the family of all positive semidefinite quadratic forms on 𝔼d there corresponds a closed convex cone 𝒬d in 𝔼(1/2)d(d+1) with apex at the origin. We describe its exposed faces and show that the families of its extreme and exposed faces coincide. Using these results, flag transitivity, neighborliness, singularity and duality properties of 𝒬d are shown. The isometries of the cone 𝒬d are characterized and we state a conjecture describing its linear symmetries. While the cone 𝒬d is far from being polyhedral, the results obtained show that it shares many properties with highly symmetric, neighborly and self dual polyhedral convex cones.
For d = 2, 3 a generic convex body in has a unique lattice packing and for d ≥ 4 at most a(d) lattice packings of maximum density, where a(d) ≥ 1 is a constant. If in a certain connectedness property holds, one may take a(d) = 1. Dually, for d = 2 a generic convex body has a unique lattice covering of minimum density and for d ≥ 3 there is a constant b(d) ≥ 1 such that it has at most b(d) such coverings.
Recent results on extremum properties of the density of lattice packings of smooth convex bodies and balls extend and refine Voronoĭ’s classical criterion for balls. This article treats in more detail the special case of lattice packings and coverings with circular discs. The aim is to determine those lattices for which the densities of the corresponding packings and coverings with circular discs, and certain products and quotients thereof, are semi-stationary, stationary, extreme, and ultra-extreme. The latter notion is a sharper version of extremality. It turns out that in all cases where solutions exist, the regular hexagonal lattices are solutions. Unexpectedly, in a few cases the square lattices and in one case special parallelogram lattices are solutions too. A further surprise is the fact that the lattices forwhich the circle packing density is extreme coincide with the lattices with ultra-extreme density. For semi-stationarity, stationarity and ultra-extremality the duality between packing and covering results breaks down. All results may be interpreted in terms of binary positive definite quadratic forms.
In this article we first prove a stability theorem for coverings in 𝔼2 by congruent solid circles: if the density of such a covering is close to its lower bound , then most of the centers of the circles are arranged in almost regular hexagonal patterns. A version of this result then is extended to coverings by geodesic discs in two-dimensional Riemannian manifolds.
Given a sufficiently differentiable convex body C in 𝔼3, the following two problems are closely related: (i) Approximation of C with respect to the Hausdorff metric, the Banach-Mazur distance and a notion of distance due to Schneider by inscribed or circumscribed convex polytopes. (ii) Covering of the boundary of C by geodesic discs with respect to suitable Riemannian metrics.
The stability result for Riemannian manifolds and the relation between approximation and covering yield rather precise information on the form of best approximating inscribed convex polytopes Pn of C with respect to the Hausdorff metric: if the number n of vertices is large, then most of the vertices are arranged in almost regular hexagonal patterns. Consequently, the majority of facets of Pn are almost regular triangles. Here ‘regular’ is meant with respect to the Riemannian metric of the second fundamental form. Similar results hold for circumscribed polytopes and also for the Banach-Mazur distance and Schneider's notion of distance.
The surface topography of red blood cells (RBCs) was investigated under near-physiological conditions using atomic force microscopy (AFM). An immobilization protocol was established where RBCs are coupled via molecular bonds of the membrane glycoproteins to wheat germ agglutinin (WGA), which is covalently and flexibly tethered to the support. This results in a tight but non-invasive attachment of the cells. Using tapping-mode AFM, which is known as gentle imaging mode and therefore most appropriate for soft biological samples like erythrocytes, it was possible to resolve membrane skeleton structures without major distortions or deformations of the cell surface. Significant differences in the morphology of RBCs from healthy humans and patients with systemic lupus erythematosus (SLE) were observed on topographical images. The surface of RBCs from SLE patients showed characteristic circular-shaped holes with approx. 200 nm in diameter under physiological conditions, a possible morphological correlate to previously published changes in the SLE erythrocyte membrane.