Related papers: Illuminating and covering convex bodies
A centrally symmetric convex body is a convex compact set with non-empty interior that is symmetric about the origin. Of particular interest are those that are both smooth and strictly convex -- known here as regular symmetric bodies --…
We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps. The cohomology theory is introduced with enough generality to be applicable to other questions in set theory.…
Illuminating the surface of a convex body with parallel beams of light in a given direction generates a shadow region. We prove sharp regularity results for the boundary of this shadow in every direction of illumination. Moreover,…
We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a…
In this note, we propose some open problems and questions about bounded convex domains in $\mathbb C^N$, specifically about visibility and iteration theory.
We propose a new shape-based modeling technique for applications in imaging problems. Given a collection of shape priors (a shape dictionary), we define our problem as choosing the right dictionary elements and geometrically composing them…
We consider the problem of camouflaging for bodies with specular surface in the framework of geometric optics. The index of visibility introduced in [Plakhov 2017] measures the mean deviation of light rays incident on the body's surface. We…
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…
The Gauss image problem for convex bodies asks for the existence of a convex body that "links" two given measures on the unit sphere in a certain way. We treat here a corresponding question for pseudo-cones, that is, for unbounded closed…
A detailed combinatorial analysis of planar convex lattice polygonal lines is presented. This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained.
There are many open problems and some mysteries connected to the realizations of the associahedra as convex polytopes. In this note, we describe three -- concerning special realizations with the vertices on a sphere, the space of all…
It is well-known since the time of the Greeks that two disjoint circles in the plane have four common tangent lines. Cappell et al. proved a generalization of this fact for properly separated strictly convex bodies in higher dimensions. We…
In his classical work, W. Blaschke proved that a convex body whose shadow boundaries are flat for every direction of parallel illumination must be an ellipsoid. An extension recently proposed by I. Gonzalez-Garc\'ia, J. Jer\'onimo-Castro,…
A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory. The method is illustrated by the examples possessing the structure typical of many problems in applied mathematics. Good numerical…
We present a different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations we prove several new identities, and simplify the proofs of several known identities. Some open problems are discussed towards…
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
A well-known conjecture states that the Whitney numbers of the second kind of a geometric lattice (simple matroid) are logarithmically concave. We show this conjecture to be equivalent to proving an upper bound on the number of new copoints…
A mathematical analogy between paraxial optics with two circular polarizations of light in a defocusing Kerr medium with positive dispersion, binary Bose-Einstein condensates of cold atoms in the phase separation regime, and hydrodynamics…
Ptychography is a technique widely used in microscopy for achieving high-resolution imaging. This method relies on computational processing of images gathered from diffraction patterns produced by several partial illuminations of a sample.…