The number of double-normals in space (Q331381)

Let \(V\) be a set of \(n\) points in \(\mathbb{R}^{d}\). One says that two points \(p,q \in V\) form a double-normal pair, if the set \(V\) lies between two parallel hyperplanes \(H_{p}\) and \(H_{q}\) that pass through \(p\) and \(q\), and that they are orthogonal to the segment \(pq\). A double normal pair \(p,q\) is called strict, if \(V \cap (H_{p} \cup H_{q}) = \{p,q\}\). Using this one may consider the (strict) double-normal graph on \(V\). We say that \(G\) is the (strict) double-normal graph of \(V\) if \(V\) is the vertex set of \(G\) and two points from \(V\) are connected by an edge iff they form a (strict) double-normal pair. In the paper the author studies what is the maximum number \(N_{d}(n)\) of double-normal pairs in a set of \(n\) points in \(\mathbb{R}^{d}\), and analogously what is the maximum number \(N{'}_{d}(n)\) of strict double-normal pairs. In terms of double-normal graphs, this question can be formulated as follows: what is the maximum number of edges in a (strict) double-normal graph of \(n\) vertices in \(\mathbb{R}^{d}\). It follows from the celebrated Erdős-Stone theorem that \[ N_{d}(n) = \frac{1}{2}(1- 1/k)n^{2} + O(n^{2}) \] for a suitable integer \(k=k(d)\), where \(k\) is the largest integer such that for any \(r\in \mathbb{N}\) we can find a double-normal graph in \(\mathbb{R}^{d}\) that has a complete \(k\)-partite graph \(K_{k}(r)\) with sizes of parts equal to \(r\) as a subgraph. An analogous statement can be obtained for \(N{'}_{d}(n)\), namely \[ N{'}_{d}(n) = \frac{1}{2}(1- 1/k')n^{2} + O(n^{2}) \] for a suitable integer \(k' = k'(d)\). It is worth pointing out that \(N_{d}(n) \geq N{'}_{d}(n)\) and \(k \geq k'\). Thus to find the asymptotic of \(N_{d}(n)\) in case \(N_{d}(n)\) being quadratic, one needs to determine \(k\). The same holds for \(N{'}_{d}(n)\). In order to approach this problem the author, firstly, improves a result due to \textit{J. Pach} and \textit{K. J. Swanepoel} [Mathematika 61, No. 1, 259--272 (2015; Zbl 1311.52016)]. Theorem 1. Let \(m\geq 2\) and suppose that there exist \(m\) points \(p_{1},\dots,p_{m} \in \mathbb{R}^{d}\), and \(m\) unit vectors \(u_{1},\dots,u_{m} \in \mathbb{R}^{d}\) such that for all triples of distinct \(i,j,k\) the angle \(\angle (p_{i},p_{j},p_{k})\) is acute, and we have the following inequalities for the inner products \[ \langle u_{i}, p_{i} - p_{j} \rangle < \langle u_{i}, p_{k} - p_{j} \rangle < \langle u_{i}, p_{j} - p_{i} \rangle . \] Then, for any \(N \in \mathbb{N}\), there exists a strict double-normal graph in \(\mathbb{R}^{d+m}\) containing a complete \(m\)-partite \(K_{m}(N)\). In particular, \(k'(d+m) \geq m\). Let \(D(m)\) be the minimal dimension \(d\) in which \(m\) points forming only non-obtuse angles exists, and let \(D'(m)\) be the analogous function for acute angles. The main result of the paper is the following. Theorem 2. One has \[ k(d) + D(k(d)) \leq d, \] \[ k'(d) + D'(k'(d)) \geq d. \] Combining Theorem 1 and Theorem 2 the author observes that \[ k'(4) = k(4) = 2, k'(5) = k(5) = 3, k'(7) = k(7) = 4. \]