Question

Let D be convex and open in R² and let f:D to R be convex .let [a,b] ×[c,d] be a closed rectangle contained in D ,where a,b,c,d element R with a<b and c<d .prove that there exists k element of R such that ,

|f(x,y)-f(u,v)|<k (|x-u|+|y-v|) for all ((x,y);(u,v)) element of

[a,b]×[c,d].

Answer 1

Since D is convex and open, $f$ being a convex function satisfies local Lipschitz condition and hence continuous. So in particular $f$ is continuous on the rectangle. Hence it is continuous on a compact set. So is the function $((x,y),(u,v))\mapsto\frac{|f(x,y)-f(u,v)|}{|x-u|+|y-v|}.$ Hence continuous image being compact, its image is compact in $\mathbb{R}$ and hence bounded. We take $k$ as the bound.