Neoconjugate norms and a generalized ladder problem

In the first quadrant of the plane, we consider the L-shaped region R = {(x, y) | x ≤ a or y ≤ b}. For p ≠ 0 and for a vector (x,y) in the plane, define ||x, y)||p = (xp + yp) 1/p, called the p-length of the vector. We find the maximum L such that a segment of fixed p-length L can go around the corner within R. This is ||(a,b)||q where 1/q - 1/p = 1. This is equal to the p-length of the segment across the corner in R from (0,0) to (a, b). With p = 2 and q = 2/3, this solves a classical optimization problem of calculus. Our analysis uses envelopes of families of curves, as well as elementary inequality methods. © Applied Probability Trust 2014.