A great circle arrangement is a finite collection of great circles on a sphere, together with the subdivision
into spherical polygons determined by those
circles (Fejes Tóth 1987). For 
distinct circles with no three concurrent, the number
of regions is 
. The first circle gives two regions, and the
th
circle adds 
regions for 
.
For such arrangements on the unit sphere, let 
and 
denote the smallest and largest cell areas. Bhowmick (2026)
proved, uniformly over the arrangements, that
 |
where 
denotes big-O notation. He also proved 
for sufficiently large 
. These estimates improve bounds in two problems of
Fejes Tóth but do not settle the conjectured extremal constants.
