Bounds on variable-length compound jumps

Understand

In Euclidean space there is a trivial upper bound on the maximum length of a compound "walk" built up of variable-length jumps, and a considerably less trivial lower bound on its minimum length.

• The existence of this non-trivial lower bound is intimately connected to the triangle inequalities, and the more general "polygon inequalities".
• Moving beyond Euclidean space, when a modified version of these bounds is applied in "rapidity space" they provide upper and lower bounds on the relativistic composition of velocities.
• Similarly, when applied to "transfer matrices" these bounds place constraints either (in a scattering context) on transmission and reflection coefficients, or (in a parametric excitation context) on particle production.