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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