Let \((M, \textrm{d})\) be a metric space. A regular ellipsoid in \((M, \textrm{d})\) based on focal points x and y with radius r, denoted by \(\textrm{E}_{M,\textrm{d}}(x,y; r) \) , is the set \( \{ z\in M:\ \textrm{d}(x, z) + \textrm{d}(z, y) \le \textrm{d} (x,y) + 2 r \} \) . We call \(\textrm{E}_{M, \textrm{d}}(x,y; 0) \) the geodesic closed interval with endpoints \(x, y \in M\) . A set \(X\subseteq M\) is convex in \((M, \textrm{d})\) provided \(\textrm{E}_{M, \textrm{d}}(x,y; 0) \subseteq X\) for all \(x,y\in X\) . What do these regular ellipsoids and convex sets in \((M, \textrm{d})\) look like? For any positive integer k, what is the largest size of a set \(X\subseteq M\) such that the convex hull of any \(Y\in \left( {\begin{array}{c}X\\ k\end{array}}\right) \) is disjoint from \(X\setminus Y\) ? What is the maximum size of a set \(X\subseteq M\) such that, for any two points \(x,y\in X\) , the geodesic closed interval with endpoints x, y is always disjoint from \(X\setminus \{x,y\}\) ? We address the above-mentioned questions for both Manhattan metric spaces and Chebyshev metric spaces, which are metric spaces induced by the \( l _1\) -norm and the \( l _\infty \) -norm on finite-dimensional real vector spaces.
