Generic lower lifted facets project to a triangulation of the convex hull

Fix $x\in \mathrm{conv}(P)$. A feasible representation is a vector $({\lambda }_i{)}_{i\in P}$ with ${\lambda }_i\ge 0$, ${\sum }_i{\lambda }_i=1$, and ${\sum }_i{\lambda }_i{p}_i=x$. Its formal height is $H(\lambda )={\sum }_i{\lambda }_i{h}_i$. If its support has more than $d+1$ sites, the vectors $(p_i,1)\in {\mathbb{R}}^{d+1}$ are linearly dependent, so there is a nonzero vector $({\alpha }_i)$ on that support with ${\sum }_i{\alpha }_i=0$ and ${\sum }_i{\alpha }_i{p}_i=0$. Such a vector has both positive and negative entries. Choose one of the two directions $\lambda +t\alpha$ that does not increase $H$ and increase $\mathrm{\mid }t\mathrm{\mid }$ until a coefficient first becomes zero. This preserves feasibility and reduces support. Repeating finitely gives a feasible representation of support at most $d+1$ with no greater height.

There are finitely many supports of size at most $d+1$. For any affinely independent support, its coefficients representing $x$ are unique; any dependent support reduces by step 1.1. Therefore the minimum formal height among feasible representations is attained among a finite nonempty set of basic supports.

Exclude temporarily the finite union of convex hulls of subsets of at most $d$ sites. For an $x$ outside this union, choose a minimizing support of smallest cardinality; it has exactly $d+1$ sites, is affinely independent, and has strictly positive coefficients ${\lambda }_i$. Indeed, a dependent support reduces by step 1.1 without increasing its minimal height, contradicting minimal cardinality. For any other site $q$, write its unique affine coordinates $q={\sum }_i{\alpha }_i{p}_i$, ${\sum }_i{\alpha }_i=1$. For sufficiently small positive $t$, replacing ${\lambda }_i$ by ${\lambda }_i-t{\alpha }_i$ and adding coefficient $t$ at $q$ remains feasible. Minimality gives $h_q-{\sum }_i{\alpha }_i{h}_i\ge 0$. Hence the hyperplane through the lifted support sites lies below every lifted site and its projected simplex contains $x$.

The set excluded in step 3.1 lies in a finite union of proper affine subspaces. A line segment in any open ball of $\mathrm{int}\mathrm{conv}(P)$ can be chosen in a direction not parallel to any of these subspaces; each subspace then removes at most one point of the segment. Thus the complement is dense in the interior. The interior is dense in the whole convex hull: join any hull point to a fixed interior point, and every point of the joining segment other than the hull endpoint is interior by convexity. There are finitely many $d$-tuples of sites, hence finitely many lower simplices; their projected simplices are closed. Step 3.1 puts a dense subset of $\mathrm{conv}(P)$ in their union, so that finite union covers the entire convex hull.

If a lower hyperplane contained at least $d+2$ lifted sites, its projected face would contain a nondegenerate $d$-tuple and an additional site, contradicting the nonzero determinant from [F3]. Thus every full-dimensional lower cell is a simplex. If two projected lower simplices have support planes $L_1,L_2$, all their vertices lie on or above both supporting planes. For a point $x$ in the intersection of their projections, the two lifted convex combinations at $x$ therefore force $L_1(x)\ge L_2(x)$ and $L_2(x)\ge L_1(x)$; hence $L_1(x)=L_2(x)$. On each simplex the corresponding plane difference has one sign at every vertex, so its zero set is a face. If $L_1=L_2$, the nonzero determinant [F3] forces the same $d+1$ vertices. Thus intersections are common faces.

The supporting inequality for a formal lift has nonnegative constant-term difference at each site; otherwise its negative constant coefficient would force a negative formal sign by [F3]. The constant terms therefore give the unperturbed weak support inequalities, and [F4] identifies them with empty power spheres.