The view-obstruction problem asks how much a closed convex body containing the origin in its interior
must be dilated about the origin
so that its translates to the half-integer lattice points meet every ray
from the origin into the open positive orthant.
For the Euclidean-ball version in 
,
let 
be the infimum of radii 
such that the closed balls centered at 
meet every such ray
(Cusick 1973, Kahlon 2026).
Kahlon (2026) reported
 |
and
 |
These are squared radii. Under the older dilation convention using diameter-one balls, the corresponding squared dilation
is 
. A positive integer direction is primitive
when its coordinates have greatest common divisor
1. Among these directions, extremality occurs only at permutations
of 
in dimension six and 
in dimension seven. The reported values depend
on exact-search certificates and require independent verification. Kahlon (2026)
reports Gemini assistance with implementing and optimizing the exact search.
Replacing Euclidean balls with axis-aligned cubes gives a view-obstruction formulation of the lonely runner conjecture. The cube and ball thresholds use different distance norms and are not interchangeable (Kahlon 2026).
