73 research outputs found
Reconstructing a Simple Polytope from its Graph
Blind and Mani (1987) proved that the entire combinatorial structure (the
vertex-facet incidences) of a simple convex polytope is determined by its
abstract graph. Their proof is not constructive. Kalai (1988) found a short,
elegant, and algorithmic proof of that result. However, his algorithm has
always exponential running time. We show that the problem to reconstruct the
vertex-facet incidences of a simple polytope P from its graph can be formulated
as a combinatorial optimization problem that is strongly dual to the problem of
finding an abstract objective function on P (i.e., a shelling order of the
facets of the dual polytope of P). Thereby, we derive polynomial certificates
for both the vertex-facet incidences as well as for the abstract objective
functions in terms of the graph of P. The paper is a variation on joint work
with Michael Joswig and Friederike Koerner (2001).Comment: 14 page
Helly-type problems
In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals
On the Monotone Upper Bound Problem
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of
vertices on a strictly-increasing edge-path on a simple d-polytope with n
facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n)
provided by McMullen's (1970) Upper Bound Theorem is tight, where M_{ubt}(d,n)
is the number of vertices of a dual-to-cyclic d-polytope with n facets.
It was recently shown that the upper bound M(d,n)<=M_{ubt}(d,n) holds with
equality for small dimensions (d<=4: Pfeifle, 2003) and for small corank
(n<=d+2: G\"artner et al., 2001). Here we prove that it is not tight in
general: In dimension d=6 a polytope with n=9 facets can have M_{ubt}(6,9)=30
vertices, but not more than 26 <= M(6,9) <= 29 vertices can lie on a
strictly-increasing edge-path.
The proof involves classification results about neighborly polytopes, Kalai's
(1988) concept of abstract objective functions, the Holt-Klee conditions
(1998), explicit enumeration, Welzl's (2001) extended Gale diagrams, randomized
generation of instances, as well as non-realizability proofs via a version of
the Farkas lemma.Comment: 15 pages; 6 figure
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