1. CMB 2009 (vol 53 pp. 3)
|A Combinatorial Reciprocity Theorem for Hyperplane Arrangements|
Given a nonnegative integer $m$ and a finite collection $\mathcal A$ of linear forms on $\mathcal Q^d$, the arrangement of affine hyperplanes in $\mathcal Q^d$ defined by the equations $\alpha(x) = k$ for $\alpha \in \mathcal A$ and integers $k \in [-m, m]$ is denoted by $\mathcal A^m$. It is proved that the coefficients of the characteristic polynomial of $\mathcal A^m$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law.
2. CMB 2009 (vol 52 pp. 342)
|On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width |
The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions $3$, $4$, $5$ and $6$.
Keywords:almost smooth convex body, convex body of constant width, weakly neighbourly antipodal convex polytope, Illumination Conjecture, X-ray number, X-ray Conjecture
Categories:52A20, 52A37, 52C17, 52C35