1. ## Fun easy problem

Here is a problem I came up with that is extremely easy, but when I posted it on another forum it took a while for someone to answer and the answer was much more complex than what was needed. Let's see if MHF has better mathematicians....I have faith!

Question: Find the largest $\displaystyle n$ in $\displaystyle \mathbb{R}^n$ such that every k-cell (denoted $\displaystyle E$) has finitely many points $\displaystyle p,q\in{E}$ such that $\displaystyle d(q,p)=\text{diam}\left(E\right)$. Prove that this is the largest $\displaystyle n$.

2. What are diam and k-cell ?

3. Originally Posted by Moo
What are diam and k-cell ?
In a metric space

$\displaystyle {\rm{diam}}(E)=\sup_{a,b \in E}d(a,b)$,

that is it is the diameter of the set $\displaystyle E$ (where $\displaystyle d(a,b)$ is the "distance from $\displaystyle a$ to $\displaystyle b$, that is $\displaystyle d$ is the metric).

Now my guess is that $\displaystyle k$ and $\displaystyle n$ are supposed to be the same thing, but who am I to say.

What supprises me is that with the level of clarity of the statement of this problem Mathstud28 got any solutions let alone one correct but more complicated ( ) than what he thought was needed.

CB

4. Originally Posted by Moo
What are diam and k-cell ?
In my books they don't say n-cell...hmm...maybe they do. but anyways CaptainBlack said what $\displaystyle \text{diam}(E)$, and a k-cell is defined to be: if $\displaystyle a_i<b_i~~{i=1,\cdots{k}}$ then a k-cell is the set of all points $\displaystyle \bold{p}=(p_1,\cdots{p_k})$ in $\displaystyle \mathbb{R}^k$ whose coordinates satisfy the inequalities $\displaystyle a_i\leqslant{a_i}\leqslant{b_i}~~1\leqslant{i}\leq slant{k}$