Stefani_problem_stefani_problem -

∑i=1k+1fi2=(∑i=1kfi2)+fk+12sum from i equals 1 to k plus 1 of f sub i squared equals open paren sum from i equals 1 to k of f sub i squared close paren plus f sub k plus 1 end-sub squared Substitute the inductive hypothesis:

A[i,j]+A[k,l]≤A[i,l]+A[k,j]cap A open bracket i comma j close bracket plus cap A open bracket k comma l close bracket is less than or equal to cap A open bracket i comma l close bracket plus cap A open bracket k comma j close bracket stefani_problem_stefani_problem

Look into Monge Arrays to see how these "Gnome" properties allow for faster shortest-path algorithms in geometric graphs. ∑i=1k+1fi2=(∑i=1kfi2)+fk+12sum from i equals 1 to k plus