Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles
  and Multiscale Inequalities   We define a discrete Menger-type curvature of d+2 points in a real separable
Hilbert space H by an appropriate scaling of the squared volume of the
corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors
d-regular measure on H by integrating the discrete curvature according to the
product measure. The aim of this work, continued in a subsequent paper, is to
estimate multiscale least squares approximations of such measures by the
Menger-type curvature. More formally, we show that the continuous d-dimensional
Menger-type curvature is comparable to the ``Jones-type flatness''. The latter
quantity adds up scaled errors of approximations of a measure by d-planes at
different scales and locations, and is commonly used to characterize uniform
rectifiability. We thus obtain a characterization of uniform rectifiability by
using the Menger-type curvature. In the current paper (part I) we control the
continuous Menger-type curvature of an Ahlfors d-regular measure by its
Jones-type flatness.

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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio