Positivity of $\Delta$-genera for connected polarized demi-normal schemes

In this paper, we show that the $\Delta$-genus $\Delta(X,\mathcal{L})\ge 0$ for any connected polarized demi-normal scheme $(X,\mathcal{L})$. As a direct corollary, we obtain $\Delta(X,K_X+\Lambda)\ge 0$ for any Gorenstein stable log scheme $(X,\Lambda)$, which is an inequality of Noether's type.