On the Number of Guard Edges of a Polygon.
On the number of intersections of two polygons
We study the maximum possible number of intersections of the boundaries of a simple -gon with a simple -gon in the plane for . To determine the number is quite easy and known when or is even but still remains open for and both odd. We improve (for ) the easy upper bound to and obtain exact bounds for
On the Number of Minimal 1-Steiner Trees.
On the number of minimal pairs of compact convex sets that are not translates of one another
Let [A,B] be the family of pairs of compact convex sets equivalent to (A,B). We prove that the cardinality of the set of minimal pairs in [A,B] that are not translates of one another is either 1 or greater than ℵ₀.
On the Number of Views of Polyhedral Terrains.
On the Openness of Continuous Averagings.
On the Openness of the Barycentre Map.
On the packing densities of superballs and other bodies.
On the perimeter of the intersection of congruent disks.
On the points of multiplicity of monotone operators
On the radius of a set in a Hilbert space
On the radius of convergence of an entire function in a normed space
On the rank of a topological convexity
On the Realization of Convex Polytopes, Euler's Formula and Möbius Functions (Short Communication)
On the Realization of Convex Polytopes, Euler's Formula and Möbius Functions
On the reduction of pairs of bounded closed convex sets
Let X be a Hausdorff topological vector space. For nonempty bounded closed convex sets A,B,C,D ⊂ X we denote by A ∔ B the closure of the algebraic sum A + B, and call the pairs (A,B) and (C,D) equivalent if A ∔ D = B ∔ C. We prove two main theorems on reduction of equivalent pairs. The first theorem implies that, in a finite-dimensional space, a pair of nonempty compact convex sets with a piecewise smooth boundary and parallel tangent spaces at some boundary points is not minimal. The second theorem...
On the refinements of a polyhedral subdivision.
Let pi: P --> Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by pi (or pi-induced) and the fiber polytope of the projection: a polytope Sygma(P,pi) of dimension dim(P)-dim(Q) whose faces are in correspondence with the coherent pi-induced subdivisions (or pi-coherent subdivisions). In this paper we investigate the structure of the poset of pi-induced refinements of a pi-induced subdivision....
On the Representation and Multiplication of Basic Alternating Cycle Matrices
On the rigidity of one-dimensional systems of contraction similitudes.
On the Rockafellar theorem for -monotone multifunctions
Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let be a cyclic -monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the -subdifferential of f, .