Support Function for Set of Points in $\mathbb{R}^2$

[squenshl]

Homework Statement

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.

Homework Equations

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

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!.

 

[RUber]

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 $-|p| max_{x \in S}( |x| )$.

 

[squenshl]

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
$$\mu_S(x_1,x_2) = \begin{cases}
x_1+x_2, & \text{if} \; x_1, x_2 \geq 0 \\
x_1, & \text{if} \; x_1 \geq 0, x_2 < 0 \\
x_2, & \text{if} \; x_1 < 0, x_2 \geq 0 \\
0 & \text{otherwise}
\end{cases}.$$