Let us say I have a closed compact convex set $\mathbb{S}$ on the 2-D plane (eg: a circle). Let any point $p$ in the 2-D plane be represented by $p=(x,y)$. I define the max function over 2-D plane \begin{align} f(p)&=\max{(x,y)} \\ &= \begin{cases}x & \text{if $x\geq y$} \\ y & \text{if $x
Let us say, the corner points of this convex body in the all-negative quadrant are known. i.e. Let $P_w=(x_w,y_w)$ be the west-most point, $P_{sw}=(x_{sw},y_{sw})$ , the most south-west point such that it lies on $x=y$, and let $P_{s}=(x_s,y_s)$ be the south-most point. Then
- if $x_w>y_w$, then $c^{\star}=x_w$
- if $y_s>x_s$, then $c^{\star}=y_s$
- if $y_w >x_w$ and $y_s
, then $c^{\star}=x_{sw}$
How do I extend this to 3-Dimensional given the corner points of this body