Erdös distance problems in normed spaces
The investigations trace back to an old Erdös problem [\textit{P. Erdös}, Am. Math. Mon. 53, 248-250 (1946; Zbl 0060.34805)]. The author studies the problems of the maximum of unit distances, largest distances and smallest distances among \(n\) points in a two-dimensional normed space. For each norm \(|\cdot|\) on \(\mathbb{R}^2\) let \(\lambda(|\cdot|)\) be the length of the longest line-segment contained in the boundary of the unit disc \(\{x: |x|\leq 1\}\). (It holds \(0\leq \lambda(|\cdot|\leq 2\) and \(\lambda(|\cdot|)=0\) iff the norm is strictly convex and \(\lambda(|\cdot|)=2\) iff the unit disc is a parallelogram.) The exact maximum numbers of largest and smallest distances for each normed space is determined, the maximum numbers of unit distance for each normed space in which the unit sphere is not strictly convex. Some remarkable results are: 1. Let \(u(n)\) be the maximum number of occurrences of the unit distance in a set of \(n\) points in \((\mathbb{R}^2,|\cdot |)\). For \(n\geq 11\) yields -- \(c_1n\log n\leq u(n)\leq c_2n^{3/4}\), if \(\lambda(|\cdot |)=0\), -- \(u(n)= \lfloor n^2/4\rfloor\), if \(0<\lambda(|\cdot|)\leq 1\), -- \(u(n)= \lfloor(n^2+ n)/4\rfloor\), if \(1<\lambda(|\cdot|)\leq 2\). 2. Let \(l(n)\) be the maximum number of occurrences of the largest distance in a set of \(n\) points in \((\mathbb{R}^2,|\cdot|)\). Then -- \(l(n)=n\), if \(\lambda(|\cdot|)=0\), -- \(l(n)=\lfloor n^2/4\rfloor\), if \(0<\lambda(|\cdot|)\leq 1\), -- \(l(n)=\lfloor n^2/4\rfloor+1\), if \(1<\lambda(|\cdot|)< 2\), -- \(l(n)=\lfloor n^2/4\rfloor+2\), if \(\lambda(|\cdot|)=2\). 3. Let \(S(n)\) be the maximum number of occurrences of the smallest distance in a set of \(n\) points in \((\mathbb{R}^2,|\cdot|)\). Then -- \(s(n)=\lfloor 3n-\sqrt {12n-3}\rfloor\), if \(\lambda(|\cdot|)<2\), -- \(s(n)=\lfloor 4n-\sqrt {28n-12}\rfloor\), if \(\lambda(|\cdot|)=2\). The author gives also a lot of interesting lemmas and consequences, and he includes results from other papers (in particular in relation to the Euclidean plane).
scientific article; zbMATH DE number 902885