Minimal contact circuits for characteristic functions of spheres

We study the complexity of implementation of the characteristic functions of spheres by contact circuits. By the characteristic functions of the sphere with center at a vertex $\tilde\sigma=(\sigma_1,\ldots,\sigma_n)$, $\sigma_1,\ldots,\sigma_n\in\{0,1\}$, we mean the Boolean function $\varphi^{(n)}_{\tilde\sigma}(x_1,\ldots,x_n)$ which is equal to 1 on those and only those tuples of values that differ from the tuple $\tilde\sigma$ only in one digit. It is shown that the number $3n-2$ of contacts is necessary and sufficient for implementation of $\varphi^{(n)}_{\tilde\sigma}(\tilde x)$ by a contact circuit.