Jürgen Bokowski

The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes

The classification of the 1296 (simplicial) 3-spheres with nine vertices into polytopal and nonpolytopal spheres, started earlier, is completed here. It is shown that there are 1142 polytopal and 154 nonpolytopal such spheres, and a fast procedure for their construction is described.

Polytopal and nonpolytopal spheres an algorithmic approach

The convexity theory for oriented matroids, first developed by Las Vergnas [17], provides the framework for a new computational approach to the Steinitz problem [13]. We describe an algorithm which, for a given combinatorial (d − 2)-sphereS withn vertices, determines the setCd,n(S) of rankd oriented matroids withn points and face latticeS. SinceS is polytopal if and only if there is a realizableM εCd,n(S), this method together with the coordinatizability test for oriented matroids in [10] yields a decision procedure for the polytopality of a large class of spheres. As main new result we prove that there exist 431 combinatorial types of neighborly 5-polytopes with 10 vertices by establishing coordinates for 98 “doubted polytopes” in the classification of Altshuler [1]. We show that for alln ≧k + 5 ≧8 there exist simplicialk-spheres withn vertices which are non-polytopal due to the simple fact that they fail to be matroid spheres. On the other hand, we show that the 3-sphereM9639 with 9 vertices in [2] is the smallest non-polytopal matroid sphere, and non-polytopal matroidk-spheres withn vertices exist for alln ≧k + 6 ≧ 9.

On the coordinatization of oriented matroids

Several important and hard realizability problems of combinatorial geometry can be reduced to the realizability problem of oriented matroids. In this paper we describe a method to find a coordinatization for a large class of realizable cases. This algorithm has been used successfully to decide several geometric realizability problems. It is shown that all realizations found by our algorithm fulfill the isotopy property.

Neighborly 2-Manifolds with 12 Vertices

We provide a complete list of 59 orientable neighborly 2-manifolds with 12 vertices of genus 6, and we study their possible flat embeddings in Euclidean 3-space. Whereas the question of embeddability remains open in its general form, we obtain several properties of the embedding (polyhedral realization) under the assumption that it does exist: 1.The order of the geometrical automorphism group of any polyhedral realization would not exceed 2. 2.The polyhedral realization would not be obtainable via the Schlegel diagram of any 4-polytope; moreover, none of our orientable neighborly 2-manifolds with 12 vertices can be found within of the 2-skeleton of any 4-polytope. 3.The polyhedral realization would not allow a tetrahedral subdivision without inserting new vertices. By using a weaker version of the manifold property, we obtain neighborly polyhedra with 2nvertices for everyn?3.

Integral representations of quermassintegrals and Bonnesen-style inequalities

First we give a simplified proof of known integral representations of Minkowskis quermassintegrals of a convex body. These representations are closely related to Simons generalized Minkowski integral formula. An application of the integral representations yields sharps inequalities involving three quermassintegrals the circum-radius. It is shown that equality occurs only for balls.

On the Finding of Final Polynomials

Final polynomials have been used to prove non-representability for oriented matroids, i.e. to decide whether geometric embeddings of combinatorial structures exist. They received more attention when Dress and Sturmfels, independently, pointed out that non-representable oriented matroids always possess a final polynomial as a consequence of an appropriate real version of Hilberts Nullstellensatz. We discuss the more difficult problem of determining such final polynomials algorithmically. We introduce the notion of bi-quadratic final polynomials, and we show that finding them is equivalent to solving an LP-Problem. We apply a new theorem about symmetric oriented matroids to a series of cases of geometrical interest.

Altshuler's sphere M10-425- is not polytopal

There is still no algorithm to decide in reasonable time whether a combinatorial sphere is polytopal or not. A short method of proof for showing that a combinatorial sphere is not polytopal is described in the case of a 3-sphere with 10 vertices (Altshulers M 425 10 ).

Nonrealizability proofs in computational geometry

This paper deals with a class of computational problems in real algebraic geometry. We introduce the concept of final polynomials as a systematic approach to prove nonrealizability for oriented matroids and combinatorial geometries.Hilberts Nullstellensatz and its real analogue imply that an abstract geometric object is either realizable or it admits a final polynomial. This duality has first been applied by Bokowski in the study of convex polytopes [7] and [11], but in these papers the resulting final polynomials were given without their derivations.It is the objective of the present paper to fill that gap and to describe an algorithm for constructing final polynomials for a large class of nonrealizable chirotopes. We resolve a problem posed in [10] by proving that not every realizable simplicial chirotope admits a solvability sequence. This result shows that there is no easy combinatorial method for proving nonrealizability and thus justifies our final polynomial approach.

On combinatorial and affine automorphisms of polytopes

We disprove the longstanding conjecture that every combinatorial automorphism of the boundary complex of a convex polytope in euclidean spaceEd can be realised by an affine transformation ofEd.

A geometric realization without self-intersections does exist for Dyck's regular map

Even in our decade there is still an extensive search for analogues of the Platonic solids. In a recent paper Schulte and Wills [13] discussed properties of Dycks regular map of genus 3 and gave polyhedral realizations for it allowing self-intersections. This paper disproves their conjecture in showing that there is a geometric polyhedral realization (without self-intersections) of Dycks regular map {3, 8}6 already in Euclidean 3-space. We describe the shape of this new regular polyhedron.