Valeriu Soltan

Geometric methods and optimization problems

I. Nonclassical Variational Calculus. II. Median Problems in Location Science. III. Minimum Convex Partitions of Polygonal Domains.

The Erdos-Szekeres problem on points in convex position – a survey

In 1935 Erdős and Szekeres proved that for any integer n ≥ 3 there exists a smallest positive integer N(n) such that any set of at least N(n) points in general position in the plane contains n points that are the vertices of a convex n-gon. They also posed the problem to determine the value of N(n) and conjectured that N(n) = 2n−2 + 1 for all n ≥ 3. Despite the efforts of many mathematicians, the Erdős-Szekeres problem is still far from being solved. This paper surveys the known results and questions related to the Erdős-Szekeres problem in the plane and higher dimensions, as well as its generalizations for the cases of families of convex bodies and the abstract convexity setting.

Minimum dissection of a rectilinear polygon with arbitrary holes into rectangles

In this paper, the problem of dissecting a plane rectilinear polygon with arbitrary (possibly, degenerate) holes into a minimum number of rectangles is shown to be solvable inO(n3/2 logn) time. This fact disproves a famous assertion about the NP-hardness of the minimum rectangular dissection problem for rectilinear polygons with point holes.

Antipodality properties of finite sets in Euclidean space

This is a survey of known results and still open problems on antipodal properties of finite sets in Euclidean space. The exposition follows historical lines and takes into consideration both metric and affine aspects.

Lower bounds for the numbers of antipodal pairs and strictly antipodal pairs of vertices in a convex polytope

The greatest lower bounds for the numbers of antipodal pairs and strictly antipodal pairs of vertices in a convexd-polytope withn vertices are determined.

Minimum Convex Partition of a Polygon with Holes by Cuts in Given Directions

Let \({\cal F}\) be a given family of directions in the plane. The problem of partitioning a planar polygon P with holes into a minimum number of convex polygons by cuts in the directions of \({\cal F}\) is proved to be NP-hard if \(|{\cal F}| \ge 3\) and it is shown to admit a polynomial-time algorithm if \(|{\cal F}| \le 2\) .

Minimum dissection of rectilinear polygon with arbitrary holes into rectangles

In this paper, the problem of dissecting a plane rectilinear polygon with arbitrary (possibly, degenerate) holes into a minimum number of rectangles is shown to be solvable inO(n3/2 logn) time. This fact disproves a famous assertion about the NP-hardness of the minimum rectangular dissection problem for rectilinear polygons with point holes.

On Grünbaum's Conjecture about Inner Illumination of Convex Bodies

Abstract. In 1964 Grünbaum conjectured that any primitive set illuminating from within a convex body in Ed , d ≥ 3 , has at most 2d points. This was confirmed by V. Soltan in 1995 for the case d = 3 . Here we give a negative answer to Grünbaums conjecture for all d ≥ 4 , by constructing a convex body K ⊂ Ed with primitive illuminating sets of an arbitrarily large cardinality.

Convex Quadrics and Their Characterizations by Means of Plane Sections

Ellipses and ellipsoids form a well-established special class of convex surfaces, primarily due to a wide range of their applications in various mathematical disciplines. The present survey deals with a natural extension of this class to that of convex quadrics. It contains a classification of convex quadrics of the Euclidean space R n and describes, in terms of plane quadric sections, their various characteristic properties among all convex hypersurfaces of R n , possibly unbounded.

Choquet simplexes in finite dimension — A survey

This survey covers various geometric results related to Choquet simplexes in the Euclidean space Ed; it describes the known properties of Choquet simplexes and marks still open problems.