On affine subspaces that illuminate a convex set.
On hyperplanes and semispaces in max–min convex geometry
The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.
On mappings preserving a family of star bodies.
On smooth points of boundaries of open sets
The notions of smooth points of the boundary of an open set and α(·) intrinsically paraconvex sets are introduced. It is shown that for an α(·) intrinsically paraconvex open set the set of smooth points is a dense -set of the boundary.
On starshapedness of the union of closed sets in
On the Hadwiger numbers of centrally symmetric starlike disks.
On the Number of Guard Edges of a Polygon.
On the rank of a topological convexity
On the shortest curve which meets all the lines which meet a circle
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 ,...
Polygonizations of Point Sets in the Plane.
Semi-monotone sets
A coordinate cone in is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is an open bounded subset of , definable in an o-minimal structure over the reals, such that its intersection with any translation of any coordinate cone is connected. This notion can be viewed as a generalization of convexity. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone...
Séparation forte -branlante de deux convexes
Sets Expressible as Unions of Staircase -Convex Polygons
Let and be fixed, , , and let be a simply connected orthogonal polygon in the plane. For lies in a staircase -convex orthogonal polygon in if and only if every two points of see each other via staircase -paths in . This leads to a characterization for those sets expressible as a union of staircase -convex polygons , .
Shaking compact sets.
Shortest Watchman Routes in Simple Polygons.
Stabilité isopérimétrique
Staircase Kernels for Orthogonal d-Polytopes.
Starshapedness in convexity spaces
Streckenförmigkeit - eine weitere Verallgemeinerung der Sternförmigkeit