Two exercises about hypermetric spaces 
Take $S$ to be the collection of all subsets of $\{1,\dots,n\}$. If $x, y$ are in $S$, define $d(x,y)$ as the number of elements of the symmetric difference $x\triangle y$.
Exercise 2.1. Show that $(S,d)$ is a metric space. Moreover, $S$ satisfies the following stronger version of the triangle inequality: if $x_1,\dots,x_k$ are elements of $S$ and $b_1,\dots,b_k$ are integers that add up to $1$, then $$\sum_{i,j=1}^k b_ib_jd(x_i,x_j)\le0$$
Exercise 2.2. Define the distance between two points $(x,y,z)$ and $(x',y',z')$ in a three-dimensional space as the biggest of the numbers $|x-x'|$, $|y-y'|$, $|z-z'|$. Show that this distance is not hypermetric.

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These two exercises are about hypermetric. For the first one, I believe that induction for k should work. But I don't know what's special about the metric, and cannot prove the inequality for $k=4$. As for the second exercise, it seems that the inequality might fail when $k=4$, but I have no idea how to find those four points. Any help is appreciated, thank you!
 A: For the first exercise: write the metric $d$ as the sum of semi-metrics
$$d_r(A,B)=\begin{cases}1,\quad \text{ if }r\in A\triangle B \\ 0 \quad\text{otherwise}\end{cases}$$
For each $r$, we have
$$\sum_{i,j} b_ib_jd_r(A_i,A_j)
= \sum_{r\in A_i,\ r\notin A_j} b_ib_j+ \sum_{r\notin A_i,\ r\in A_j} b_ib_j \\= 2 \sum_{r\in A_i}b_i \sum_{r\notin A_j} b_j = 2\sum_{r\in A_i}b_i\left(1-\sum_{r\in A_i}b_i\right)\le 0 
$$
because $N(1-N)\le 0$ for every integer $N$. Then sum over $r$.

For the second one, I don't have a conceptual explanation, but a computer search brought up this example: take five points with coordinates given by the columns of the matrix
$$ \begin{pmatrix}    1& 1& 1& 2& 0\\    0& 2& 2& 1& 1\\    0& 1& 0& 1& 0\\    \end{pmatrix}   $$
and check that the sum in the hypermetric inequality is $1>0$.
I don't have a counterexample with $4$ points; it's been more than a year since I looked for them, but I believe my search did not turn up anything. 

For more on hypermetrics, I can offer my blog posts Pentagrams and hypermetrics and Polygonal inequalities: beyond the triangle.
