# max length or size of a convex set

I want to maximize $||x-y||$ with $x$ and $y$ in $C$ where $C$ is the intersection of some discs. We assume the intersection is nonempty, and closed. I am thinking, how to formulate it as a semidefinite programmimg problem? Does anyone know how?

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For future reference, you can edit your question. There is no need to post as a new one. – Willie Wong Oct 3 '10 at 11:59

To compute the maximum distance between to given Arcs $A_1$ and $A_2$ let $x_i\in A_i$ be two points such that $d(x_1,x_2)=\sup\{d(x,y)|x\in A_1,y\in A_2\}$. (Existence follows from compactness). If you take a unit speed parametrization $\gamma_1,\gamma_2$ of the arcs, and times $t_i$ with $\gamma_i(t_i)=x_i$. Assuming, that $x_1$ is not the endpoint of the arc $A_1$, then the function $\mathbb{R}\rightarrow \mathbb{R} \qquad t\mapsto d(\gamma_1(t),x_2)$ obtains a maximum at $t$, so its differential must be $0$ at $t_1$. But this is just the scalar product of $\gamma'_1(t_1)$ and the gradient of $d(-,x_2)$ at $\gamma(t)$, which can be seen as the direction of the line connecting $x_1$ and $x_2$.
If there are $n$ discs, how many "sides" can the intersection have? If the answer grows exponentially with $n$, your procedure - while perfectly correct - may be impractical. Also, I'm not sure I understand the assertion about computing the maximum distance between points on two given arcs (nor how "minimizing points" are relevant to a maximization problem). – Gerry Myerson Oct 3 '10 at 22:41