Triangles in convex distance planes
The author studies triangles in two-dimensional vector spaces with a \textit{convex distance function} (or \textit{gauge}), which is a distance given by the Minkowski functional of a (possibly non-symmetric with respect to the origin) convex body containing the origin as an interior point. Notions of equilateral and acute triangles are extended to this context. The \textit{radius} of a given set \(A\) is defined to be infimum of the radius of the disks containing \(A\), and a disc of minimum radius containing \(A\) is said to be a \textit{minimal enclosing disc}. A triangle is said to be \textit{equilateral} if all of its sides have the same radius, and \textit{weakly acute} if each side has a minimal enclosing disc which does not contain the respective opposite vertex as an interior point. The author proves that every equilateral triangle is weakly acute. Also, the author investigates \textit{central sets} and \textit{bisectors}, which are, roughly speaking, sets of equidistant points of a given set. It is proved that the central set of a weakly acute triangle is non-empty, and that the bisector of an equilateral triangle is also non-empty.
scientific article; zbMATH DE number 6971515