Polytopes and arrangements: Diameter and curvature

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Abstract

We introduce a continuous analogue of the Hirsch conjecture and a discrete analogue of the result of Dedieu, Malajovich and Shub. We prove a continuous analogue of the result of Holt and Klee, namely, we construct a family of polytopes which attain the conjectured order of the largest total curvature.

Section snippets

Continuous analogue of the conjecture of Hirsch

By analogy with the conjecture of Hirsch, we conjecture that the order of the largest total curvature of the central path associated to a polytope is the number of inequalities defining the polytope. By analogy with a result of Dedieu, Malajovich and Shub, we conjecture that the average diameter of a bounded cell of a simple arrangement is less than the dimension. We prove a continuous analogue of the result of Holt–Klee, namely, we construct a family of polytopes which attain the conjectured

Discrete analogue of the result of Dedieu, Malajovich, Shub

Let A be a simple arrangement formed by m hyperplanes in dimension n. We recall that an arrangement is called simple if mn+1 and any n hyperplanes intersect at a unique distinct point. Since A is simple, the number of bounded cells, i.e., bounded connected components of the complement to the hyperplanes, of A is I=m-1n. Let λc(P) denote the total curvature of the primal central path corresponding to min{cTx:xP}. Following the approach of Dedieu et al. [4], let λc(A) denote the average value

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Proposition 3.1

If the conjecture of Hirsch holds, then ΔA(m,n)n+2nm-1.

Proof

Let mi denote the number of hyperplanes of A which are non-redundant for the description of a bounded cell Pi. If the conjecture of Hirsch holds, we have δ(Pi)mi-n. It implies δ(A)i=1i=I(mi-n)I=i=1i=ImiI-n.Since a facet belongs to at most 2 cells, the sum of mi for i=1,,I is less than twice the number of bounded facets of A. As a bounded facet induced by a hyperplane H of A corresponds to a bounded cell of the (n-1)-dimensional simple

Acknowledgments

Research supported by an NSERC Discovery grant, by a MITACS grant and by the Canada Research Chair program.

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