4
$\begingroup$

Consider an edge $e$ of a simplex $C$. If we increase the length of $e$ while keeping the length of other edges constant, the width of $C$ increases in some directions and decreases in other directions. Does the mean width of $C$ increase? The answer is yes for triangles, since the mean width of a triangle is proportional to its perimeter. What about simplexes of higher dimension?

Improve this question
$\endgroup$
3
  • $\begingroup$ This is vague as stated, but one precise question is: If we pick a side, fix its midpoint, move the two vertices on that side further away from each other, and fix all other vertices, can the mean width increase in that process? $\endgroup$
    – user44143
    Commented Sep 20, 2022 at 0:37
  • $\begingroup$ Your question is different from mine : the lengths of other edges change in the process. Up to isomorphism, the matrice of distances between vertices uniquely determines a simplex, and my question is precise. $\endgroup$
    – Pluviophile
    Commented Sep 20, 2022 at 2:18
  • 1
    $\begingroup$ I'd find that question more clearly asked as: "If two simplices have the same graphs of distances between vertices, with the exception of one edge that differs between them, can the simplex with the larger edge have smaller mean width?" $\endgroup$
    – user44143
    Commented Sep 20, 2022 at 3:16

1 Answer 1

Reset to default
1
$\begingroup$

Yes, the mean width is monotone with respect to the length of any edge of a tetrahedron. This follows from these two facts:

  1. The mean width of a 3-dimensional polyhedron is proportional to $\sum_i \ell_i \theta_i$, where $\ell_i$ are the edge lengths and $\theta_i$ are the exterior dihedral angles.
  2. The Schlaefli formula: $\frac{\partial}{\partial\ell_i}\sum_j \ell_j \theta_j = \theta_i$.
Improve this answer
$\endgroup$
2
  • $\begingroup$ Could you provide a reference on 1.? $\endgroup$
    – Pluviophile
    Commented Jan 14, 2023 at 3:56
  • 1
    $\begingroup$ Usually this is stated in a broader context of quermassintegrals: the average volume of projections to $k$-dimensional linear subspaces is proportional to a certain mixed volume, which is a coefficient in the Steiner formula. I can mention two books: Santalo, Integral geometry and geometric probability, and Schneider, Convex bodies: the Brunn-Minkowski theory. $\endgroup$
    – Ivan Izmestiev
    Commented Jan 14, 2023 at 6:01

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .