8 problems

Let $a$ be the parameter in the continuous $p$-hub location problem, and let Case 1 and Case 2 denote the two cases distinguished by the conditions $\Delta_1\geq\Delta_2$ and…

The network-reduction strategy may eliminate facilities and either abandon the associated demand nodes or reassign them to facilities that remain open. The complexity-cost savings…

Let $C_n$ denote the facility-location objective for the random point set $X$, and let $r_p$ be the radius associated with each point $p\in X$. The displayed analysis applies Talag…

The facility-location objective $C_n$ is analyzed using Talagrand's inequality, yielding a concentration interval of length $O\left(\sqrt{\sum_{p\in X}r_p^2}\right)$, where $X$ is…

In a two-dimensional space, consider partitioning the plane into service regions and measuring the relevant spatial cost associated with each region. Gersho's hexagonal optimality…

Let $Q$ be a convex body in the plane and let $\kappa\geq 1$. Define … and let $\Delta(Q)$ denote the diameter of $Q$. For $\kappa=1$, the relevant ratio is $\mu_Q^*/\Delta(Q)$; fo…

Let $Q$ range over all convex bodies in the plane, and let $\mu_Q^*$ be the minimum average Euclidean distance from a point in the plane to points of $Q$. Let $\Delta(Q)$ denote th…

Carmi–Har-Peled–Katz conjecture. The constants satisfy