Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometry of a Set and its Random covers   Let $E$ be a bounded open subset of $\mathbb{R}^n$. We study the following
questions: For i.i.d. samples $X_1, \dots, X_N$ drawn uniformly from $E$, what
is the probability that $\cup_i \mathbf{B}(X_i, \delta)$, the union of
$\delta$-balls centered at $X_i$, covers $E$? And how does the probability
depend on sample size $N$ and the radius of balls $\delta$? We present
geometric conditions of $E$ under which we derive lower bounds to this
probability. These lower bounds tend to $1$ as a function of $\exp{(-\delta^n
N)}$.
  The basic tool that we use to derive the lower bounds is a good partition of
$E$, i.e., one whose partition elements have diameters that are uniformly
bounded from above and have volumes that are uniformly bounded from below. We
show that if $E^c$, the complement of $E$, has positive reach then we can
construct a good partition of $E$. This partition is motivated by the Whitney
decomposition of $E$. On the other hand, we identify a class of bounded open
subsets of $\mathbb{R}^n$ that do not satisfy this positive reach condition but
do have good partitions.
  In 2D when $E^c\subset \mathbb{R}^2$ does not have positive reach, we show
that the mutliscale flat norm can be used to approximate $E$ with a set that
has a good partition under certain conditions. In this case, we provide a lower
bound on the probability that the union of the balls \emph{almost} covers $E$.

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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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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: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: eess