The symmetric wedge over a facet projects its vertex-edge graph onto the base

Let $P=\mathrm{Hpoly}(a,b)\subseteq\mathbb R^d$ be bounded with $b_i>0$ and fix a row $k$. The symmetric wedge over row $k$ is the polytope $W\subseteq\mathbb R^{d+1}$ cut out by the $n$ rows $(a_i,\,[i=k])$ with the same right-hand sides and one extra row $(a_k,-1)$ with right-hand side $b_k$; its fibre over $x\in P$ is $|t|\le b_k-\langle a_k,x\rangle$. Then the coordinate projection $\pi(x,t)=x$ sends vertices of $W$ to vertices of $P$ and sends every edge of $W$ either to an edge of $P$ or to a single point.

This is the Klee--Walkup $d$-step wedge in the symmetric normalization used by Santos (Lemma 2.4 in the polar). Vertices of $W$ lie on one of the two roofs (otherwise the vertical direction is a tight-row kernel vector), each roof is an affine copy of $P$, and an edge not inside a roof has all common tight rows with zero last coordinate, so its projection is a point. The one-sided version of this argument is the accepted platform proof of Hirsch.balanced_hpoly_transfer.