Interpolation and extrapolation of smooth functions by linear operators.
Measure and Helly's Intersection Theorem for Convex Sets
Let be a uniformly bounded collection of compact convex sets in ℝ ⁿ. Katchalski extended Helly’s theorem by proving for finite ℱ that dim (⋂ ℱ) ≥ d, 0 ≤ d ≤ n, if and only if the intersection of any f(n,d) elements has dimension at least d where f(n,0) = n+1 = f(n,n) and f(n,d) = maxn+1,2n-2d+2 for 1 ≤ d ≤ n-1. An equivalent statement of Katchalski’s result for finite ℱ is that there exists δ > 0 such that the intersection of any f(n,d) elements of ℱ contains a d-dimensional ball of measure...
Note On H-Convex Functions
Note on H-convex functions.
On Bárány's theorems of Carathéodory and Helly type
The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows:...
On order and metric convexities in
On point sets fixing a convex body from within.
On starshapedness of the union of closed sets in
On the estension of Lipschitz functions with respect to two Hilbert norms and two Lipschitz conditions
Optimal bounds for the colored Tverberg problem
We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.
Polyhedral Line Transversals in Space.
Polynomial selections and separation by polynomials
K. Nikodem and the present author proved in [3] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems on separation of an n-convex function from an n-concave function by a polynomial of degree at most n and a stability result of Hyers-Ulam type for polynomials.
Proof of Grünbaum's Conjecture on Common Transversals for Translates.
Reconstructing dimensions of intersections of convex sets.
Remark on the Helly Number for Strongly Convex Sets on Riemannian Manifolds.
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 , .
Some intersection and generation properties of convex sets
Stufenkonvexität in der Geometrie
The maximum piercing number for some classes of convex sets with the -property.
Theorems on the Existence of Separating Surfaces.