The height of an edge in a 3-polytope is the maximum degree of the two vertices and two faces incident with . In 1940, Lebesgue proved that every 3-polytope without so called pyramidal edges has an edge with . In 1995, this upper bound was improved to 10 by Avgustinovich and Borodin. Recently, we improved it to 9 and constructed a 3-polytope without pyramidal edges satisfying for each .
The purpose of this paper is to prove that every 3-polytope without pyramidal edges has an edge with .
In different terms, this means that every plane quadrangulation without a face incident with three vertices of degree 3 has a face incident with a vertex of degree at most 8, which is tight.