Isosceles sets in Euclidean spaces of dimension nine to twenty-two
Abstract
A finite set of points in ℝd is isosceles if every three of its points form an isosceles triangle. Erdős asked in 1946 for the largest size iso(d) of an isosceles set in ℝd; the answer was known for d≤8. Let g(d) and M(d) be the largest sizes of a two-distance set and of a spherical two-distance set in ℝd. We prove that iso(d)=max{g(d),M(d)+1,M(d−1)+3} for every d≥2, which reduces the problem to these two classical ones. With the recently determined values of g(d) for 9≤ d≤22 we obtain iso(d)=binomd+12+1 for 9≤ d≤21 and iso(22)=276, and we show that the extremal set is unique: for d≤21 it is the set of midpoints of the edges of a regular simplex together with its centroid, and for d=22 it is the McLaughlin configuration together with its centre. For 10≤ d≤22 every isosceles set with more than binom d2+3 points is a two-distance set or a spherical two-distance set together with its centre. The proofs start from Blokhuis' decomposition of isosceles sets into mutually orthogonal two-distance sets. Counting leaves one configuration open: a two-distance set with binomd+12 points in a hyperplane, the largest number allowed by Blokhuis' bound, together with two points that are mirror images of each other. We exclude it by showing that a two-distance set T attaining Blokhuis' bound in ℝn is unisolvent for quadratic polynomials and that, if T lies on two concentric spheres, then apart from two exceptional cases its two layers are spherical $2$-designs whose sizes are determined by the radii. We also show that the squared distances of such a set are in ratio (k−1):k with (2k−1)2≤ n+1 if n≥9. Hence there is none for 9≤ n≤23, so that g(23)≤299 and iso(23)=max{g(23),278}≤299.
Related
- Dimension Thresholds for Relative-Error Representations of Finite Metrics in Pseudo-Euclidean Spaces
- The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture
- Exact cost of the Euclidean Maximum Traveling Salesman Problem in one dimension
- Full-radius dimension-free maximal inequalities for discrete Euclidean balls
- Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
BibTeX
@misc{yi2026isosceles,
title = {{Isosceles sets in Euclidean spaces of dimension nine to twenty-two}},
author = {Hyunjun Yi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05421},
eprint = {2611.05421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05421},
}
RIS
TY - UNPB
TI - Isosceles sets in Euclidean spaces of dimension nine to twenty-two
AU - Hyunjun Yi
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.05421
UR - https://arxiv.org/abs/2611.05421
AB - A finite set of points in ℝ^d is isosceles if every three of its points form an isosceles triangle. Erdős asked in 1946 for the largest size iso(d) of an isosceles set in ℝ^d; the answer was known for d≤8. Let g(d) and M(d) be the largest sizes of a two-distance set and of a spherical two-distance set in ℝ^d. We prove that iso(d)=max{g(d),M(d)+1,M(d−1)+3} for every d≥2, which reduces the problem to these two classical ones. With the recently determined values of g(d) for 9≤ d≤22 we obtain iso(d)=binomd+12+1 for 9≤ d≤21 and iso(22)=276, and we show that the extremal set is unique: for d≤21 it is the set of midpoints of the edges of a regular simplex together with its centroid, and for d=22 it is the McLaughlin configuration together with its centre. For 10≤ d≤22 every isosceles set with more than binom d2+3 points is a two-distance set or a spherical two-distance set together with its centre. The proofs start from Blokhuis' decomposition of isosceles sets into mutually orthogonal two-distance sets. Counting leaves one configuration open: a two-distance set with binomd+12 points in a hyperplane, the largest number allowed by Blokhuis' bound, together with two points that are mirror images of each other. We exclude it by showing that a two-distance set T attaining Blokhuis' bound in ℝ^n is unisolvent for quadratic polynomials and that, if T lies on two concentric spheres, then apart from two exceptional cases its two layers are spherical $2$-designs whose sizes are determined by the radii. We also show that the squared distances of such a set are in ratio (k−1):k with (2k−1)^2≤ n+1 if n≥9. Hence there is none for 9≤ n≤23, so that g(23)≤299 and iso(23)=max{g(23),278}≤299.
ER -