Every forward sector has nonempty interior

Suppose $A_x$ has empty ambient interior. If $d=1$, S1 supplies a strict successor $y\mathrm{\ne }x$; convexity puts the nondegenerate interval between $x$ and $y$ in $A_x$, contradicting empty interior. Thus take $d\ge 2$. The affine hull of $A_x$ has dimension at most $d-1$. If its dimension were at most $d-2$, [F2] would give two distinct supporting hyperplanes at $x$, contradicting S2. Hence its affine hull is a hyperplane $H$; this $H$ supports $A_x$ at every point, including $x$.

By S1 choose $y$ with $x\prec \prec y$. Then $y\in A_x$ and $x\mathrm{\notin }{A}_y$. Transitivity gives $A_y\subseteq A_x\subseteq H$. If $A_y$ had affine dimension at most $d-2$, [F2] would again give multiple ambient supports at $y$, contrary to S2. Thus $\mathrm{aff}{A}_y=H$.

Choose $w\in {\mathrm{ri}}_H{A}_y$ by [F3]. The segment from $w$ to $x$ is contained in $H\cap {\mathrm{\Gamma }}_i$ by convexity of both sets, starts in $A_y$, and ends outside $A_y$. Its intersection with $A_y$ is closed by relative closedness [F1], so it has a last point $z$ before $x$. Points immediately after $z$ on this segment lie outside $A_y$ and, because $A_y$ is closed in the open set ${\mathrm{\Gamma }}_i$, outside ${\overline{A_y}}^H$. Thus $z\in {\mathrm{\partial }}_H{\overline{A_y}}^H$. The support statement [F2] supplies a nonzero functional on $H$ whose zero hyperplane supports $A_y$ at $z$.

Extend this functional to the ambient Euclidean space by taking its normal vector in the direction space of $H$. Its zero hyperplane contains the within-$H$ support and is not $H$ itself. Since $z\in A_y$, transitivity gives $A_z\subseteq A_y$, so the extended plane supports $A_z$ at $z$; the plane $H$ also supports $A_z$. These are distinct, contradicting S2 at $z$. Therefore $A_x$ has nonempty ambient interior.