Convex bodies of constant width and the Apollonian metric (Q957530)

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)].