Gale's straightness criterion

The Geradheitskriterium of Gale describes a condition at the vertex sets of facets of a cyclic polytope . It goes back to the mathematician David Gale .

One consequence of Gale's straightness criterion is that two cyclic polytopes of equal dimensions with the same number of vertices are combinatorially equivalent . So one can speak of the cyclic d-polytope with n vertices.

definition

Let be the set of vertices of a cyclic polytope and be a set of d vertices of the polytope. ${\displaystyle V\!\,}$ ${\displaystyle C(d,n)\!\,}$ ${\displaystyle V_{d}\!\,}$

The convex hull of is then a facet if and only if every pair of two corners is separated by an even number of corners on the moment curve . ${\displaystyle V_{d}\!\,}$ ${\displaystyle V\setminus V_{d}\!\,}$ ${\displaystyle V_{d}\!\,}$

example

${\displaystyle C(3,6)\!\,}$is a cyclic polytope in dimension 3 with 6 corners. Its corners are numbered according to their order on the torque curve. is a set consisting of three corners of the polytope. ${\displaystyle V_{3}\!\,=\{0,3,4\}}$

${\displaystyle V\setminus V_{3}\!\,}$consists of . ${\displaystyle \{1,2,5\}\!\,}$

• The corners 1 and 2 are separated by 0 corners on the moment curve.
• The corners 1 and 5 are separated from the two corners 3 and 4 on the moment curve.
• The corners 2 and 5 are separated from the two corners 3 and 4 on the moment curve.

${\displaystyle \mathrm {conv} (V_{3})\!\,}$is thus a facet of . ${\displaystyle C(3,6)\!\,}$

literature

• Branko Grünbaum with Volker Kaibel, Victor Klee, Günter M. Ziegler : Convex Polytopes. 2nd edition, Springer 2003, ISBN 0-387-00424-6 (first from Grünbaum 1967).