## Deformed Products and Maximal Shadows of Polytopes (1996)

Venue: | ADVANCES IN DISCRETE AND COMPUTATIONAL GEOMETRY, AMER. MATH. SOC., PROVIDENCE, CONTEMPORARY MATHEMATICS 223 |

Citations: | 29 - 1 self |

### BibTeX

@INPROCEEDINGS{Amenta96deformedproducts,

author = {Nina Amenta and Günter M. Ziegler},

title = {Deformed Products and Maximal Shadows of Polytopes},

booktitle = {ADVANCES IN DISCRETE AND COMPUTATIONAL GEOMETRY, AMER. MATH. SOC., PROVIDENCE, CONTEMPORARY MATHEMATICS 223},

year = {1996},

pages = {57--90},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with "many pivots," starting with the famous Klee-Minty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for d-dimensional simple polytopes with at most n facets: ffl the maximal number of vertices on an increasing path, ffl the maximal number of vertices on a "greedy" greatest increase path, and ffl the maximal number of vertices of a 2-dimensional projection. This, equivalently, provides good estimates for the worst-case behaviour of the simplex algorithm on linear programs with these parameters with the worst-possible, the greatest increase, and the shadow vertex pivot rules. The bounds on the maximal number of vertices on an increasing path or a greatest increase path unify and slightly improve a number of known results. One bound on the maximal number of vertices of a 2-dimensional projection is new: we show ...