ühekauguslik

Ühekauguslik describes a configuration of points in a metric space where the distance between any two distinct points is the same constant. The term in Estonian is formed from kaugus, meaning distance, with the prefix ü- meaning one or single. In geometry it is commonly called an equidistant set; in finite-dimensional Euclidean spaces it corresponds to the vertices of a regular simplex.

In R^d, a finite ühekauguslik set can contain at most d+1 points; when there are d+1 points, the set forms a regular simplex in that dimension. The common distance between any two vertices is a fixed c > 0. A key property is that the vectors from the centroid to the vertices have equal length and are symmetrically arranged, so the configuration exhibits high symmetry.

Examples include an equilateral triangle in the plane (d = 2) and a regular tetrahedron in three-dimensional space (d = 3). In higher dimensions, the corresponding figures are regular simplices with d+1 vertices. Such configurations are often used as canonical examples of highly symmetric point sets and serve as models in various geometric constructions.

Applications and related concepts: ühekauguslik sets are studied in discrete and computational geometry, design theory, and coding theory, where questions about symmetry, distance, and optimality arise. In Euclidean spaces of finite dimension, the maximum size is limited to d+1, while in infinite-dimensional spaces equidistant sets of larger or even infinite size may exist. The concept also connects to the study of regular simplices and their geometric and algebraic properties.