An example of complete alternance in multidimensional case

On a simplex in $n$-dimensional Euclidean space we define a function $H(x)$ with a value at the point $x = (x_1, . . . , x_n)$ equal to the harmonic mean of the numbers $x_1, . . . , x_n$. We consider a problem of uniform approximation of function $H(x)$ on a simplex with linear functions. We find the only solution to the problem. It has a complete alternance. The existence of complete alternance guarantees the strong uniqueness of the solution. We find an exact constant of strong uniqueness.