Convex bodies of constant width and the Apollonian metric
Main results of the paper under review concern characterization of convex bodies of constant width in terms of the Apollonian metric: Theorem 1.1. Let \( W \) be a regular convex body of constant width and \( D \) be its compliment in the one point compactification \(\overline{\mathbb{R}^n} \) of \(\mathbb{R}^n \) endowed by the Apollonian metric. Let \( a,b \in \partial D \) and \( \gamma\) be a circular arc joining \( a \) and \( b \) in \( D \). Then \( \gamma \) is a pseudogeodesic line if and only if it is orthogonal to \( \partial D \). Any point in \( D \) can be joined to \( \infty \in D \) by a unique pseudogeodesic line. Theorem 1.2. Let \( W \) be a convex body of a constant width and \( D \) be as above. Then any Apollonian isometry \( f: D \rightarrow \overline{\mathbb{R}^n} \) is restriction of a Möbius map. The last theorem implies that two convex bodies of constant width are similar if and only if their compliments in \( \overline{\mathbb{R}^n} \) are isometric in the Apollonian metric. The authors note that the Apollonian metric of a domain \( D \) is conformal at every point if and only if \( D \) is a ball. Reviewer's remark: The Theorem 1.2. verifies a conjecture of \textit{A. F. Beardon} formulated in [Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), New York, NY: Springer, 91--108 (1998; Zbl 0890.30030)].
scientific article; zbMATH DE number 5374764