A new characterization of convex plates of constant width

✍ Scribed by Endre Makai; Horst Martini

• Publisher: Springer
• Year: 1990
• Tongue: English
• Weight: 510 KB
• Volume: 34
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00147325

✦ Synopsis

A convex plate D c R 2 of diameter 1 is of constant width 1 if and only if any two perpendicular intersecting chords have total length => 1.

. Let D c R", n > 2, be a convex body of diameter 1. We say that D has the property (P) if any n mutually perpendicular chords, having a common point, have total length > 1. For n = 2, this property has been introduced by Schmitz [4]. He has shown that a circle as well as a Reuleaux triangle of diameter 1 have this property (even with strict inequality for non-degenerate chords), and further he stated that only plates of constant width can have property (P). We shall continue these considerations with the following statements.

THEOREM 1. A convex plate D c R 2 of diameter 1 has property (P) if and only if it is of constant width 1. Moreover, if D has constant width 1, in property (P) we have strict inequality for non-deoenerate chords.

PROPOSITION. Let a convex body D c R", n > 2, of diameter 1 satisfy the property (P) (for non-deoenerate chords). Then D is of constant width 1.

There remains the open question of whether for n > 3 we have an analogous equivalence. (One sees easily that the positive answer for R "+ 1 implies the same for R". Namely, D c R" can be induced in a convex body D' c R" + 1 of constant width 1 (cf. Chakerian and Groemer [1]), and by Lemma 1 we may consider chords of D' with a common endpoint, one parallel to the (n + 1)st basis vector and of length 0.) Anyway, a ball of diameter 1 in R" Satisfies property (P). To see this, we may clearly restrict ourselves to the case when the common point of the chords lies on the boundary, in which case i=1 i=1