Claims

../

2025_05_07_clemen_dumitrescu_liu: Equally spaced points on a short arc of a unit circle, together with the center, determine n-1 distances with multiplicities n-1, ..., 1 for every n, so the characterization by lines and circles fails.

2025_12_27_alexeev: The set (0,0), (1,0), (0,1), (0,-1) determines three distances with multiplicities 3, 2, 1 over unordered pairs and lies on no line and no circle, refuting the characterization at n = 4.

---