Consider smooth closed space curve (parametrised by its arc length) in 3 dimensional space.Does there exist atleast one point at which both the curvature & torsion attain extremum values?



Space curves and torsion
Imagine a 2D curve defined as follows: $x(t) := \kappa(s), y(t) := \tau(s)$;  

if it were true that a space curve always contains a point where curvature and torsion are simultaneously extremal, then the associated 2D curve (as defined above), would always contain a corner of its bounding box. 


Even if infinite curvatures and torsions are allowed, counterexamples can easily be constructed: take a space curve, whose associated 2D curve as defined previously, is an archimedian spiral. 


