Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Separating hyperplane theorem for non-disjoint sets

  1. Aug 4, 2014 #1
    Consider the sets ##X:= \{x\in\mathbb R^2: \enspace ||x-(-1,0)||_2 \leq 1\}## (a ball) and ##Y:=co\{(0,-1), (0,1), (1,0)\}## (a triangle).

    Both ##X## and ##Y## are compact and convex, but they aren't disjoint: ##X\cap Y = \{(0,0)\}##. Since they aren't disjoint, the most common separating hyperplane/Hahn-Banach theorems don't directly apply. However, the y-axis is a hyperplane which separates them in a weak sense.

    Can anybody point me to source for a separating hyperplane theorem that covers this example? Ideally, I'm looking for something that isn't restricted to finite dimensions.

    Thanks!
     
  2. jcsd
  3. Aug 4, 2014 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Can't you just apply Hahn-Banach to the interiors?
     
  4. Aug 4, 2014 #3
    Oh jeez. Now I'm embarrassed.

    Thank you, micromass!

    Can we strike this one from the records? :P
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook