       Re: Distance from point to set

• To: mathgroup at smc.vnet.net
• Subject: [mg55314] Re: Distance from point to set
• From: Peter Pein <petsie at arcor.de>
• Date: Sat, 19 Mar 2005 04:45:15 -0500 (EST)
• References: <d1eblq\$ek7\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Piotr Kowalski wrote:
> Hello,
>
> I would like to compute distance d(x,A) from a point 'x' to a set 'A',
> (all in R^n, where n=2 or n=3) that is:
>
>    d(x,A) = min ||x - a|| (forall a in A)
>    where: n=2 or n=3,
>           x is point in R^n, A is subset of R^n
>           ||  || is norm (euclidean, max, etc).
>
> Can I find Mathematica function or package for such problem ?
>
> P. Kowalski
>
Hello Piotr,

Mathematica has got a builtin Norm[]:

In:= (* genrating testing data *)
x = {1, 2, 3}; A = Table[Random[Integer, {-5, 10}], {5}, {3}]
Out=
{{-5, -3, 8}, {4, -1, 2}, {-5, -4, 6}, {-5, 10, 10}, {-3, 10, 5}}
In:= (* definition of the distance *)
d[el_, set_, p_:Infinity] := Min[(Norm[el - #1, p] & ) /@ set]
(* you can use 2 in place of Infinity to get euclidian norm as default *)
In:= d[x, A]
Out= 3
In:= (* the same function works for R^2*)
d[{1, 1}, {{-1, 1}, {2, 2}, {2, 1}}, 2]
Out= 1

--
Peter Pein
Berlin

```

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