The relative distance of J. D. Pryce in \({\mathbb{R}}^ 3_{\infty}\) (Q1122078)

Let E be any Banach space. A path C in E is a mapping \(t\to x(t)\) defined on a compact interval \(a\leq t\leq b\) in R and piecewise smooth in the norm topology. \textit{J. D. Pryce} [SIAM J. Numer. Anal. 21, 202-215 (1984; Zbl 0558.65036)], has defined the relative distance \(\rho (A_ 0,A_ 1)\) between two nonzero \(A_ 0,A_ 1\) in E as \(\rho (A_ 0,A_ 1)=\inf dist(A_ 1,A_ 2,C)\) over all paths from \(A_ 0\) to \(A_ 1\) and not passing through 0, where \(dist(A_ 0,A_ 1,C)=\int_{C}\frac{\| dx\|}{\| x\|}.\) He proved that \(\rho\) is a metric in \(E\setminus \{0\}\) and established its usefulness in error theory. For the specific paths in the space \(R^ 3_{\infty}\) the questions of calculation of \(\rho\) and of existence of the shortest path between two points are studied. In case \(E=R^ 2_{\infty}\) these questions have been studied by the author earlier [SIAM J. Numer. Anal. 23, 1295-1302 (1986; Zbl 0614.65011)].