Only ellipsoids have caustics.
Operations between sets in geometry
An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in -dimensional Euclidean space . It is proved that if , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, covariant, and associative if and only if it is addition for some . It is also demonstrated that if ,...
Optimal isometries for a pair of compact convex subsets of ℝⁿ
In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for metrics for all p ≥ 2 and the symmetric difference metric.
Optimality conditions for maximizers of the information divergence from an exponential family
The information divergence of a probability measure from an exponential family over a finite set is defined as infimum of the divergences of from subject to . All directional derivatives of the divergence from are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for to be a maximizer of the divergence from are presented, including new ones when is not projectable to .
Parallelbeleuchtung konvexer Körper mit glatten Rändern
Partitioning Euclidean Space.
Permanence of moment estimates for p-products of convex bodies
It is shown that two inequalities concerning second and fourth moments of isotropic normalized convex bodies in ℝⁿ are permanent under forming p-products. These inequalities are connected with a concentration of mass property as well as with a central limit property. An essential tool are certain monotonicity properties of the Γ-function.
Perturbations and approximations of support functions of convex bodies.
Piecewise polynomial functions, convex polytopes and enumerative geometry
Points extrémaux d'un convexe compact de Rn appartenant à un réseau
Points of infinitely many normals to convex surfaces.
Polarity of embedded and circumscribed ellipsoids.
Polyhedra inscribed and circumscribed to convex bodies.
Polyhedral approximation of convex sets with an application to large deviation probability theory.
Preassigning the Boundary of Diametrically-Complete Sets.
Presssigning the Shape for Bodies of Constant Width.
Projecting the surface measure of the sphere of
Properties of distance functions on convex surfaces and applications
If is a convex surface in a Euclidean space, then the squared intrinsic distance function is DC (d.c., delta-convex) on in the only natural extrinsic sense. An analogous result holds for the squared distance function from a closed set . Applications concerning -boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.
Quantitative Steinitz's Theorems with Applications to Multifingered Grasping.
Quelques questions soulevées par le cône barrière d'un convexe