# Ordinary hyperspheres and spherical curves

## TL;DR

This paper investigates the minimal and maximal counts of special hyperspheres determined by points in real space, extending classical geometric problems to higher dimensions and spherical settings.

## Contribution

It solves the $d$-dimensional spherical analogue of the Dirac–Motzkin conjecture and the orchard problem for even dimensions, providing new bounds in higher-dimensional geometry.

## Key findings

- Minimum number of ordinary hyperspheres determined
- Maximum number of $(d+2)$-point hyperspheres in even dimensions
- Extension of classical geometric conjectures to higher dimensions

## Abstract

An ordinary hypersphere of a set of points in real $d$-space, where no $d+1$ points lie on a $(d-2)$-sphere or a $(d-2)$-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly $d+1$ points of the set. Similarly, a $(d+2)$-point hypersphere of such a set is one that contains exactly $d+2$ points of the set. We find the minimum number of ordinary hyperspheres, solving the $d$-dimensional spherical analogue of the Dirac--Motzkin conjecture for $d \geqslant 3$. We also find the maximum number of $(d+2)$-point hyperspheres in even dimensions, solving the $d$-dimensional spherical analogue of the orchard problem for even $d \geqslant 4$.