Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector,
and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v\_0, ... , v\_n$, where $n\ge m$). 


Let's further assume that 


* $dist(v\_i,H) = 0,\ i\in [0,k]\ \wedge\ dist(v\_i,H) \gt 0,\ i\in[k+1,n]$
* $dist(v\_j,H') = 0,\ j\in [0,k+d]\ \wedge\ dist(v\_j,H') \gt 0,\ j\in[k+d+1,n]\ \wedge\ d\ge 1$


Here $dist(\ ,\ )$ denotes the signed distance.


**Question:** 


Is it true that under the conditions stated above, we have 
$$
 \sum\_{i=0}^{n}dist(v\_i,H)\ \gt\ \sum\_{j=0}^{n}dist(v\_j,H'),
$$  

respectively, are there counterexamples known? 


As Fedor Petrov's shows, that is not generally true, but (as a follow up question) what about
$$
 \sup\_H\sum\_{i=0}^{n}dist(v\_i,H)\ \gt\ \sum\_{j=0}^{n}dist(v\_j,H') ?
$$ 



An optimality condition for the corners of convex polytopes?
No. Take a triangle $v\_0v\_1v\_2$ on the plane, $H'$ is its side $v\_0v\_1$, $H$ is almost another side $v\_0v\_2$. Then $\sum\_{i=0}^2 {\rm dist}\, (v\_i,H')$ is just a length of altitude from $v\_2$, $\sum\_{i=0}^2 {\rm dist}\, (v\_i,H)$ is almost the length of altitude from $v\_1$, which may be less than that from $v\_2$. 


As for your new question, the answer is still negative for triangles. Assume that a median $v\_0 p$ is perpendicular to $H'=v\_0v\_1$. For any line $H$ passing through $v\_0$ and not cutting the triangle we have $\sum\_{i=0}^2 {\rm dist}\,(v\_i,H)=2\,{\rm dist}\, (p,H)\leqslant 2|v\_0p|$ and maximum is achieved for $H=H'$.


