In this short survey we want to present some of the impact Minkowski's successive minima within Convex and Discrete Geometry. Originally related volume an $o$-symmetric convex body, point out relations other functionals, as e.g., lattice enumerator or intrinsic volumes old new conjectures about them. Additionally, discuss application a version Siegel's lemma.

In this article we discuss classic theorems from Convex Geometry in the context of topological drawings and beyond. a simple drawing complete graph $$K_n$$ , any two edges share at most one point: either common vertex or point where they cross. Triangles can be viewed as convex sets. This gives link to geometry. As our main result, present generalization Kirchberger’s theorem that is purely com...

Motivated by typical questions from computational geometry (visibility and art gallery problems) and combinatorial geometry (illumination problems) we present an analogue of the Krein-Milman theorem for the class of star-shaped sets. If S ⊆ R is compact and star-shaped, we consider a fixed, nonempty, compact, and convex subset K of the convex kernel K0 = ck(S) of S, for instance K = K0 itself. ...

In this article, we study two open problems posed by Edelman & Reiner about topology of certain simplicial complexes associated with abstract convex geometries. As a special case, we concentrate on generalized convex shellings, which were introduced by Kashiwabara, Nakamura & Okamoto for their representation theorem of abstract convex geometries. We answer one of the problems affirmatively for ...

begin{abstract} In this paper, we prove some strong and weak convergence of three step random iterative scheme with errors to common random fixed points of three asymptotically nonexpansive nonself random mappings in a real uniformly convex separable Banach space. end{abstract}

An algorithm for computing the convex hull of a simple polygon is presented. Its correctness and complexity are also shown. Computational geometry Convex hull Simple polygon Algorithm