Slopes of bisectors in normed planes (Q1943355)

Let \(E\) be a two-dimensional normed space. A pair \((x_1,x_2)\) of distinct points \(x_1,x_2\in E\) is called \textbf{strict} (in \(E\)) if the unit circle \(S(1)\) contains no line segment parallel to \(x_2-x_1\). The \textit{bisector} of two points \(x_1,x_2\in E\), \(x_1\neq x_2\) is the set \[ \text{Bi}(x_1,x_2) := \{x\in E\,:\, \|x-x_1\|_E = \|x-x_2 \|_E \}, \] i.e., the set of points with equal distance to the given points. This set is homeomorphic to a line if \((x_1,x_2)\) is strict. Given a strict pair \((x_1,x_2)\) we define a vector \(e:=(x_2-x_1)/\|x_2-x_1\|_E\). Moreover, let \(u\in S(1)\) be so that the unit disc \(\bar B(1)\) has a supporting line at \(u\) parallel to \(x_2-x_1\) and \(h:\mathbb R \to \mathbb R\) be a continuous function. The first main result (Theorem 2.4) is the following: Theorem. Let \((x_1,x_2)\) be a strict pair and let \(e,u,h\) be defined as above with \(\text{Bi}(x_1,x_2)= \{h(s)e+su\,:\, s\in\mathbb R \}\). Then, for all \(s,t\in\mathbb R\) (\(s\neq t\)), \[ |h(s)-h(t)| < 2 |s-t|. \] Now, the main result of the paper (Theorem 2.15) provides sharp bounds for the derivative of the real function representing the bisector: Theorem. Let \(x_1,x_2\in E\) and \(h:\mathbb R \to (-1,1)\) be as in the aforementioned theorem. Then, for all \(s\in\mathbb R\), \[ |h'_+(s)| < 2 \quad \text{and} \quad |h'_-(s)| < 2, \] with right and left derivatives \(h'_+,h'_-\). The last part of the paper is a discussion in higher dimensions, showing that this case is clearly different from the two-dimensional one.