An optimality condition for the corners of convex polytopes?

Question

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 $$ \sup_H\sum_{i=0}^{n}dist(v_i,H)\ \gt\ \sum_{j=0}^{n}dist(v_j,H') ? $$

Answer 1

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 $H=H'$.

Answer 2

It just occured to me that the question can be answered in the affirmative way for the following reasons:

• the order relation doesn't change if the distance sum is divided by some positive constant value $c$.

• if we divide the distance sums by $n+1$, then that value equals the distance of the vertices' center of mass $R$ to the same plane.

• all local maxima of $R$'s distances to points on the convex hull of the vertices are attained in a vertex which also is a corner of the convex hull and consequently also the global maximum of the distances is attained in such a vertex.

• $R$'s distance to a plane $E$ containing a vertex $v$ that also is a corner of the convex hull attains a local maximum if $R-v$ is orthogonal to $E$.

• A vertex $v^*$ with maximal distance to $R$ lies on the boundary of the smallest hypersphere centered at $R$, wich contains all vertices and, the plane $E^*$ that contains $v^*$ and is orthogonal to $R-v^*$ consequently also is a tangent plane of that hypersphere and thus the intersection of $E^*$ with the vertices' convex hull is $0$-dimensional and identical to $v^*$.