Best approximation of a three-variable function by sum of one-variable coordinate functions

Consider the following approximation problem of a continuous function of three variables by the sum of three continuous functions of one variable: $$E(f,\mathrm{\Omega })=\inf \mathrm{\mid }\mathrm{\mid }f(x,y,z)-\phi (x)-\psi (y)-\omega (z)\mathrm{\mid }{\mathrm{\mid }}_{\mathrm{\infty }}=?$$ where $f(x,y,z)$ is a given continuous function defined on $\mathrm{\Omega }=[0,1{]}^3$ and the infimum runs over all triplets of continuous functions $\phi (x),\psi (y),\omega (z)$ defined on the unit interval $[0,1]$. In this paper, we will prove a formula to calculate the error $E(f,\mathrm{\Omega })$ under certain conditions.