Bi-Lipschitz embedding of projective metrics (Q2922924)

This paper gives a new sufficient condition for a Busemann-type projective metric on a convex subset of \(\mathbb{R}^n\) to be bi-Lipschitz equivalent to a Euclidean metric on a certain set in \(\mathbb{R}^n\). More precisely, a metric \(d\) on a convex set \(\Omega \subset \mathbb{R}^n\) is called \textit{Busemann-type projective} if there exists a positive Borel measure \(\nu\) on the space \(H\) of all \((n-1)\)-dimensional affine subspaces of \(\mathbb{R}^n\) with the following properties: NEWLINENEWLINENEWLINENEWLINE (a) \(\nu(\pi K) < \infty\) for any compact set \(K \subset \Omega\); NEWLINENEWLINENEWLINENEWLINE (b) \(d(x,y) = \nu(\pi [x,y])\) for any \(x, y \in \Omega\), where for a set \(E \subset \mathbb{R}^n\), \(\pi E\) denotes the collection of all spaces \(A \in H\) that intersect \(E\). NEWLINENEWLINENEWLINENEWLINE Further, a map \(f: X \to Y\) between the metric spaces \((X,d_X)\) and \((Y,d_Y)\) is \textit{\(\eta\)-quasisymmetric}, where \(\eta: [0,\infty) \to [0,\infty)\) is a homeomorphism, if NEWLINENEWLINENEWLINE\[NEWLINEd_Y(f(x),f(y)) \leq \eta\Big(\frac{d_X(x,y)}{d_X(x,z)}\Big) d_X(x,z)NEWLINE\]NEWLINENEWLINENEWLINE for any triple of distinct points \(x,y,z\) in \(X\). NEWLINENEWLINENEWLINENEWLINE The author proves that if \(d\) is a Busemann-type projective metric on a convex subset \(\Omega\) of \(\mathbb{R}^n\) such that the identity map on \(\Omega\) is locally \(\eta\)-quasisymmetric with respect to \(d\) and the Euclidean metric (on \(\Omega\)), then \((\Omega,d)\) is bi-Lipschitz equivalent to a certain subset \(\Omega'\) of \(\mathbb{R}^n\) with the Euclidean metric. Moreover, \(\Omega' = \mathbb{R}^n\) provided \(\Omega = \mathbb{R}^n\). The bi-Lipschitz embedding of \((\Omega,d)\) into the Euclidean space is given by the formulaNEWLINENEWLINE NEWLINE\[NEWLINEf(x) = \int_{\pi[o,x]} n(A) \text{d}\,\nu(A),\quad x \in \Omega,NEWLINE\]NEWLINE NEWLINENEWLINEwhere \(\nu\) is a Borel measure on \(H\) witnessing the Busemann-type projectivity of \(d\), \(o\) is a fixed point of \(\Omega\) and, for \(A \in H\), \(n(A)\) is the unit normal vector at \(A\) that points out the halfspace containing \(o\). The paper is concluded with two related questions.