Re: distance between sets

From: Julien Santini (santini.julien_at_wanadoo.fr)
Date: 09/18/04 

Date: Sat, 18 Sep 2004 22:42:03 +0200

> Define the distance dist(X,Y) between two sets X and Y in a metric
> space E as dist(X,Y)=inf{d(x,y):x in X, y in Y}. Let X and Y be two
> disjoint and non empty sets in R^k. Show that - if X is compact and Y
> closed - there exists x and y such that dist(A,B)=dist(a,b).
>

Note: H = Y Intersection Ball(radius=big_enough), center = some point of X).
R^k being finite dimensional, H is compact.
Now ... use the fact that the "distance" function is continuous, and that a
continuous function on a compact set reaches his maximum (and minimum) on
this set. 

--
Julien Santini

Relevant Pages

  • Re: distance between sets
    ... > Define the distance distbetween two sets X and Y in a metric ... > disjoint and non empty sets in R^k. ... Show that - if X is compact and Y ...
    (sci.math)
  • Re: separation of compact and closed set
    ... >Let K be compact and F be closed in some metric space. ... >this is a continuous function on a compact set so it attains its ... >now I just need to show its minimum is not zero. ... But a distance of 0, actually achieved, means what? ...
    (sci.math)
  • Re: The example of outer regular but not inner regular measure!
    ... If f is the continuous function with compact support, ... linear functional of f to approach to infinite. ... take a infinite sequence of neighborhood consisting fixed V whose ...
    (sci.math)
  • Re: The example of outer regular but not inner regular measure!
    ... If f is the continuous function with compact support, ... linear functional of f to approach to infinite. ... take a infinite sequence of neighborhood consisting fixed V whose ...
    (sci.math)
  • Re: A few basic functional analysis questions
    ... So it's enough to show that if m< oo and eps> 0, ... But we know there is a compact subset K of E with m(E \ ... desired sequence, which leads to the result you want. ...
    (sci.math)
Quantcast