Improved bounds for geometric permutations (Q2903522)

Let \({\mathcal K }\) be a collection of \(n\) pairwise disjoint convex bodies in \( {\mathbb R}^d \). The authors recall the definition of geometric permutation: If an oriented line meets all these bodies, then it determines a well-defined order, called geometric permutation.NEWLINENEWLINELet \( g_d (n) \) be the maximal possible number of these permutations generated by a set \({\mathcal K }\).NEWLINENEWLINEThe goal of the paper is to improve estimates of \( g_d (n) \) for \( d \geq 3\), obtained in some 20-years-old papers. So, the main results of the paper are:NEWLINENEWLINETheorem 2.5. Any collection of \( n \) pairwise disjoint convex sets in \( {\mathbb R}^3 \) admits at most \( O (n^3 \log n)\) geometric permutations.NEWLINENEWLINETheorem 3.5. Any collection of \( n \) pairwise disjoint convex sets in \( {\mathbb R}^d \), for \( d \geq 3 \), admits at most \( O (n^{2d-3} \log n)\) geometric permutations.NEWLINENEWLINEThe authors formulate also the hypothesis that the correct upper bound of \( g_d (n) \) is close to \( O(n^{d-1})\) for any \( d \geq 3 \), and that \( g_3 (n) = O(n^{3})\).