Characterization of optimal trajectories in $\mathbb {R}^3$

We characterize the set of all trajectories $\mathcal T$ of an object $t$ moving in a given corridor $Y$ that are furthest away from a family $\mathbb{S}=\{S\}$ of fixed unfriendly observers. Each observer is equipped with a convex open scanning cone $K(S)$ with vertex $S$. There are constraints on the multiplicity of covering the corridor $Y$ by the cones $K$ and on the “thickness” of the cones. In addition, pairs $S$, $S'$ for which $[S,S']\subset (K(S)\cap K(S'))$ are not allowed. The original problem $\max_{\mathcal T}\min\{ d(t,S):\ t\in \mathcal T,\ S\in \mathbb S\},$ where $d(t,S)=\|t-S\|$ for $t\in K(S)$ and $d(t,S)=+\infty$ for $t\not\in K(S)$, is reduced to the problem of finding an optimal route in a directed graph whose vertices are closed disjoint subsets (boxes) from $Y\backslash \bigcup_{S} K(S)$. Neighboring (adjacent) boxes are separated by some cone $K(S)$. An edge is a part $\mathcal {T}(S)$ of a trajectory $\mathcal T$ that connects neighboring boxes and optimally intersects the cone $K(S)$. The weight of an edge is the deviation of $S$ from $\mathcal {T}(S)$. A route is optimal if it maximizes the minimum weight.