Reprint Server

Affine Invariant Measures in Levi-Civita Vector Spaces and the Erdös Obtuse Angle Theorem


An interesting question posed by Paul Erdös 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.

M. Berz, S. Troncoso, AMS Contemporary Mathematics 596 (2013) 1-21


Click on the icon to download the corresponding file.

Download Adobe PDF version (295361 Bytes).

Go Back to the reprint server.
Go Back to the home page.

This page is maintained by Ravi Jagasia. Please contact him if there are any problems with it.