The negative gradient of a log-homogeneous cone barrier is dual feasible

Fix $y\in \mathrm{int}K$ and let $\varphi (t)=F(x+ty)$ for $t\ge 0$. The sum of an interior cone point and a cone point is interior, so $\varphi$ is defined and convex. Log-homogeneity gives $\varphi (t)=F(y+x\mathrm{/}t)-\nu \log t$ for $t>0$, which tends to $-\mathrm{\infty }$ as $t\to \mathrm{\infty }$. If $DF(x)[y]={\varphi }^{\mathrm{\prime }}(0)>0$, convexity would imply $\varphi (t)\ge \varphi (0)+t{\varphi }^{\mathrm{\prime }}(0)\to +\mathrm{\infty }$, a contradiction. Thus $DF(x)[y]\le 0$ for all interior $y$.

For any $y\in K$ and $\epsilon >0$, $y+\epsilon x\in \mathrm{int}K$: an open ball around $\epsilon x$ lies in $K$, and adding $y\in K$ keeps that ball in $K$. Step 1.1 applied to $y+\epsilon x$ and continuity of the linear functional $DF(x)$ as $\epsilon \downarrow 0$ give $DF(x)[y]\le 0$. Hence $-\mathrm{\nabla }F(x)$ pairs nonnegatively with every $y\in K$, so it lies in $K^{\ast }$ by [F1]. Differentiating $F(\alpha x)=F(x)-\nu \log \alpha$ at $\alpha =1$ gives $DF(x)[x]=-\nu$.