The vertex-face weight of edges in $3$-polytopes

The weight $w(e)$ of an edge $e$ in a $3$-polytope is the maximum degree-sum of the two vertices and two faces incident with $e$. In 1940, Lebesgue proved that each $3$-polytope without the so-called pyramidal edges has an edge $e$ with $w(e)\le21$. In 1995, this upper bound was improved to 20 by Avgustinovich and Borodin. Note that each edge of the $n$-pyramid is pyramidal and has weight $n+9$. Recently, we constructed a $3$-polytope without pyramidal edges satisfying $w(e)\ge18$ for each $e$. The purpose of this paper is to prove that each $3$-polytope without pyramidal edges has an edge $e$ with $w(e)\le18$. In other terms, this means that each plane quadrangulation without a face incident with three vertices of degree $3$ has a face with the vertex degree-sum at most $18$, which is tight.