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Bounds on variable-length compound jumps

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. Physically these are very different contexts, but mathematically there are intimate relations between these superficially very distinct systems.