The analyst's traveling salesman theorem characterizes the bounded sets in the plane that lie on a one-dimensional rectifiable set. For a dyadic
square 
, let 
be the concentric square with three times the side length
and define
 |
where the infimum is over all lines, 
is the distance from 
to 
, and 
is the side length of 
. The value 
measures how closely 
resembles a straight line at the location and scale of 
.
The theorem states that a bounded set 
is contained in a curve
of finite arc length iff
 |
where the sum is over all dyadic squares (Jones 1990). Unlike the classical traveling salesman problem, the theorem is a geometric characterization rather than an algorithm for visiting a finite list of points.
