1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Support function

  1. Oct 6, 2015 #1
    1. The problem statement, all variables and given/known data
    Let ##\left\{(x_1,x_2) \in \mathbb{R}^2: 0 \leq x_1 \leq 1 \; \text{and} \; 0 \leq x_2 \leq 1\right\}.## Find the support function ##\mu_s## for this set.

    2. Relevant equations
    We define the support function ##\mu_s: \mathbb{R}^n \rightarrow \mathbb{R} \cup \left\{-\infty\right\}## as ##\mu_s(p) = \inf\left\{p \cdot x: x \in S\right\}##.

    3. The attempt at a solution
    I know this is a square with vertices at ##(0,0)##, ##(0,1)##, ##(1,0)## and ##(1,1)##. I'll take a line that goes through ##(0,1)## and take a vector ##p## that is orthogonal to this. I get stuck after this in finding the support function

    Someone please help!!!!.
     
  2. jcsd
  3. Oct 6, 2015 #2

    RUber

    User Avatar
    Homework Helper

    I think you need to find the maximum size of a vector in S, since the infimum of the dot product of p with an element x in S will be ## -|p| max_{x \in S}( |x| )##.
     
  4. Oct 9, 2015 #3
    Thanks. Here we are basically trying to maximise ##p_1x_1+p_2x_2## subject to the constraint ##p_1 \geq 0## and ##p_2 \leq 1.## The support function is
    $$\mu_S(x_1,x_2) = \begin{cases}
    x_1+x_2, & \text{if} \; x_1, x_2 \geq 0 \\
    x_1, & \text{if} \; x_1 \geq 0, x_2 < 0 \\
    x_2, & \text{if} \; x_1 < 0, x_2 \geq 0 \\
    0 & \text{otherwise}
    \end{cases}.$$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted