On convexity proofs in location theory

It is often assumed in the facility location literature that functions of the type ø_i(x_i, y) = β_i(x_i-x)²+(y_i-y)²]^K/2 are twice differentiable. Here we point out that this is true only for certain values of K. Convexity proofs that are independent of the value of K are given.