Yu and Liu’s strong duality theorem under the time-sharing property requires the Slater condition to hold for the considered general nonconvex problem, what is satisfied for the specific application. We further extend the scope of the theorem under Ky Fan convexity which is slightly weaker than Yu&Lui’s time-sharing property.

Radon’s theorem asserts that any set S of d + 2 points in [Wd has a partition into two subsets S, and S, such that Conv(S,) rl Conv(S,) # 0, where Conv(Si) denotes the convex hull of Si. This central theorem in the theory of convexity has been extended in many directions. One of the most interesting generalizations is the following theorem, proved by Tverberg in 1966, and now considered as a cl...

The generalized Roper-Suffridge extension operator in Banach spaces is introduced. We prove that this operator preserves the starlikeness on some domains in Banach spaces and does not preserve convexity in some cases. Furthermore, the growth theorem and covering theorem of the corresponding mappings are given. Some results of Roper and Suffridge and Graham et al. in Cn are extended to Banach sp...

Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were in purely combinatorial way. This led to extensions earlier results about points and halfplanes pseudohalfplanes, including polychromatic colorings discrete Helly-type theorems pseudohalfplanes. Here we continue this line research introduce the notion convex sets such pseudohalfp...

Carathéodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus counterparts have been proved by various authors. In this paper, more advanced results in discrete geometry are shown to have also their max-plus counterparts: namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem – Sierksma’s con...