Reach (mathematics)
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"Reach" mathematics
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December 2009
)
Let
X
be a subset of
R
n
. Then the
reach
of
X
is defined as
reach
(
X
)
:=
sup
{
r
∈
R
:
∀
x
∈
R
n
∖
X
with
d
i
s
t
(
x
,
X
)
<
r
exists a unique closest point
y
∈
X
such that
d
i
s
t
(
x
,
y
)
=
d
i
s
t
(
x
,
X
)
}
.
{\displaystyle {\text{reach}}(X):=\sup\{r\in \mathbb {R} :\forall x\in \mathbb {R} ^{n}\setminus X{\text{ with }}{\rm {dist}}(x,X)<r{\text{ exists a unique closest point }}y\in X{\text{ such that }}{\rm {dist}}(x,y)={\rm {dist}}(x,X)\}.}