Let $\Gamma\subset\mathbb C^n$ be a convex polytope and let $h_\Gamma(z)=\max_{v\in\Gamma}{\rm Re}\langle z,v\rangle$ be its support function with respect to the standard scalar product on $\mathbb C^n$ obtained as the real part of the standard hermitian one.

If ${\rm d}^c=i(\bar\partial-\partial)$, $B_m$ is the unit $m$-dimensional ball and $\varkappa_m$ its $m$-dimensional Lebesgue measure, Boris Yakovlevich Kazarnovskii, in the paper entitled "On the zeros of exponential sums" (Soviet Math. Dokl. Vol. 23, (1981), no. 2, 347-351), claims that the wedge product of positive currents ${\rm dd}^c h_\Gamma \wedge ({\rm dd}^c h_{B_{2n}})^{\wedge(n-1)}$ yields a positive measure $\mu_1$ such that $\mu_1(B_{2n})$ equals the sum, as $\Delta$ runs in the set ${\mathcal B}(\Gamma,1)$ of sides of $\Gamma$, of $\varkappa_{2n-1}^{-1}vol_1(\Delta)vol_{2n-1}(K_\Delta\cap B_{2n})$, where $vol_1(\Delta)$ is the length of $\Delta$, $K_\Delta$ its dual cone and $vol_{2n-1}(K_\Delta\cap B_{2n})$ the $(2n-1)$-Lebesgue measure of $K_\Delta\cap B_{2n}$.

The paper provides a more general statement about the measure $\mu_k=({\rm dd}^c h_\Gamma)^{\wedge k} \wedge ({\rm dd}^c h_{B_{2n}})^{\wedge(n-k)}$ and, although I understand the reason why $\mu_k$ is a well defined positive measure, I am confused about the way to prove both statements.

I can prove that the value $\langle\!\langle {\rm dd}^c h_\Gamma,\varsigma\rangle\!\rangle$ of the current ${\rm dd}^c h_\Gamma$ on any $(n-1,n-1)$-test form is given by the following expression: $$ \langle\!\langle {\rm dd}^c h_\Gamma,\varsigma\rangle\!\rangle=\sum_{\Delta\in{\mathcal B}(\Gamma,1)} vol_1(\Delta)\int_{K_\Delta} \upsilon_{E_\Delta^\prime}\wedge \iota^*_\Delta\varsigma $$ where $E_\Delta$ is the line issuing from the origin and parallel to $\Delta$, $E_\Delta^\prime=i E_\Delta$, $\upsilon_{E_\Delta^\prime}$ is the volume form on $E_\Delta^\prime$ and $\iota_\Delta:K_\Delta\to\mathbb C^n$ is the inclusion mapping.

I think that the preceding representation of ${\rm dd}^c h_\Gamma$ can be of some help in proving Kazarnovskii's claims, however my computation does not seem to give the expected answer.

Could anybody help me to understand how to prove Kazarnovskii's claim? Thank you in advance.