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