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?

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

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 $n\ge m$).

lets further assume that

• $dist(v_i,H) = 0,\ i\in [0,k]\ \wedge\ dist(v_i,H) \gt 0,\ i\in[k+1,n]$

• $dist(v_j,H') = 0,\ j\in [0,k+d]\ \wedge\ dist(v_j,H') \gt 0,\ j\in[k+d+1,n]\ \wedge\ d\ge 1$

$dist(\ ,\ )$ is to be understood as the signed distance.

$$ $$

Question:

is it true, that under the conditions stated above, that

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

resp., are counter examples known?

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?