On the section of a convex polyhedron

✍ Scribed by Peter Frankl; Hiroshi Maehara; Junichiro Nakashima

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 117 KB
• Volume: 140
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(93)e0172-z

✦ Synopsis

Let P be a convex polyhedron in R s, and E be a plane cutting P. Then the section Pt=Pc~E is a convex polygon. We show a sharp inequality (the perimeter of Pe) <~ L(P), where L(P) denotes the sum of the edge-lengths of P.

For a polyhedron (or a polygon) X, L(X) denotes the sum of the edge-lengths of X. Thus if X is a polygon then L(X) is the perimeter of X.

Let P be the surface of a convex polyhedron in R 3, and E be a plane cutting P. Then the section PE = PnE is a convex polygon, see Fig. 1. We prove the following theorem.

Theorem. L (PE) < ~ L (P).

We use the following lemma.

Lemma. If X is a convex polygon contained in another convex polyffon Y, then L(X) ~ L( Y).