Affine Invariant Measures in Levi-Civita Vector Spaces and the Erd?s Obtuse Angle Theorem

摘要

An interesting question posed by Paul Erdos around 1950 pertains to the maximal number of points in n-dimensional Euclidean Space so that no subset of three points can be picked that form an obtuse angle. An unexpected and surprising solution was presented around a decade later. Interestingly enough the solution relies in its core on properties of measures in n-dimensional space. Beyond its intuitive appeal, the question can be used as a tool to assess the complexity of general vector spaces with Euclidean-like structures and the amount of similarity to the conventional real case. We answer the question for the specific situation of non-Archimedean Levi-Civita vector spaces and show that they behave in the same manner as in the real case. To this end, we develop a Lebesgue measure in these spaces that is invariant under affine transformations and satisfies commonly expected properties of Lebesgue measures, and in particular a substitution rule based on Jacobians of transformations. Using the tools from this measure theory, we will show that the Obtuse Angle Theorem also holds on the non-Archimedean Levi-Civita vector spaces.