  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 an
application, we obtain $\Delta(X,I(K_X+\Lambda))\ge 0$ for any KSBA stable log
scheme $(X,\Lambda)$, where $I$ is the Cartier index of $K_X+\Lambda$. We also
construct examples of KSBA stable log schemes with $I=1$ and
$\Delta(X,K_X+\Lambda)= 0$, which shows the inequality is sharp when $I=1$.
