Contemporary Mathematics

Volume 596, 2013

Aﬃne Invariant Measures in Levi-Civita Vector Spaces and

the Erd¨ os Obtuse Angle Theorem

Martin Berz and Sebastian Troncoso

Abstract. An interesting question posed by Paul Erd¨ os 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.

1. Introduction

In order to formulate the obtuse angle problem more clearly and put it into

context, we begin with some observations about the matter at hand. First, let us

formulate it in appropriate mathematical terminology. Let V be a vector space over

a totally ordered field F . Let ( , ) denote an inner product, i.e. a function from

V

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→ F that has the common linearity properties under vector addition and scalar

multiplication on both sides. We say three points p0, p1 and p2 form an obtuse

angle at p0 if the vectors p1 − p0 and p2 − p0 have negative inner product, i.e. if

(p1 − p0,p2 − p0) 0. We say the three points form an obtuse angle if any one of

the three permutations form an obtuse angle. Furthermore, we say a set of n points

forms an obtuse angle if there are three points in the set that do so. Apparently this

algebraic notation generalizes the concept of obtuse angles in elementary geometry

and the well-known Euclidean vector spaces of Rd.

Let us now provide some perspective on the matter of point sets admitting

obtuse angles for the vector spaces Rd and the common inner product. We begin

by observing that apparently in Rd it is always possible to find sets of 2d points

that only admit non-obtuse angles, namely by merely picking the corner points of

the unit cube [0, 1]d. More specifically, because of the rotational symmetry of the

unit cube, without loss of generality we can assume p0 = (0, ..., 0); and since any of

©2013 American Mathematical Society

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