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What is the definition of distance between a point and a set of points?

  1. Mar 31, 2005 #1

    for simplicity, consider the real number space
    the distance between two points x, y (two reals) is [tex]|x - y|[/tex]
    Is there a definition of distance between a point x and a subset of R, such as an interval (a, b)?

    If there isn't any, how would you define it, such that there are some meaningful constructions?
  2. jcsd
  3. Mar 31, 2005 #2
    Maybe the distance between a point and the locus {}
  4. Mar 31, 2005 #3
    The standard definition of the distance between a point x and a set S is:

    [tex]d(x,S)=\inf\lbrace d(x,y)\colon y\in S\rbrace[/tex]

    where d is your distance function.

    Basically, the distance between a point and a set is the minimum distance between the point and every element in the set. That's not completely correct, since there may not be a minimum (which is why we use "inf" and not "min"), but that's the basic idea.
  5. Mar 31, 2005 #4
    according to the definition, the distance between 1 and (2,3) is the inf which is |1 - 2| = 1?
    If the set is an empty set, what is the distance?
  6. Mar 31, 2005 #5


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    The DEFINITION of "the distance between the point p and the set of points A" is
    "The greatest lower bound of all distances from p to each point in A"

    That is guaranteed (by the greatest lower bound property) to exist as long as A is NOT EMPTY.

    The distance from a point to the empty set is not defined.
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