Faces of intricate polytopes

Definitions and mathematical statement

For a full-dimensional convex polytope $P\subset\mathbb R^d$, let $f_i(P)$ be its number of $i$-dimensional faces. Vertices correspond to $i=0$ and facets to $i=d-1$. Count nonempty proper faces by $F(P)=\sum_{i=0}^{d-1}f_i(P)$; including the empty face and $P$ itself adds only two. For each feasible $n$, consider \[ M_d(n)=\max\{F(P):f_0(P)=f_{d-1}(P)=n\}. \] Determine the growth of this function for each fixed dimension. In particular, is $M_4(n)/n$ unbounded along feasible $n$? This asks for superlinear total face complexity despite having only $n$ vertices and $n$ facets.

A closely related four-dimensional formulation asks whether the fatness $\phi(P)=(f_1(P)+f_2(P))/(f_0(P)+f_3(P))$ is uniformly bounded over all four-polytopes, without requiring $f_0=f_3$. These formulations should keep their different input constraints explicit. In higher dimensions the source asks whether constructions from joins of polygons have the optimal order of growth. The join of $P$ and $Q$ is their convex hull after placing them in disjoint affine subspaces so that its dimension is $\dim P+\dim Q+1$.

Short English statement

Can a convex polytope have far more intermediate-dimensional faces than vertices and facets? Determine the sharp growth, especially in dimension four.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3800004 (AMR-037-0004), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Erickson - Open Problems in Computational Geometry, Faces of intricate polytopes. Original source indexed by AMR-037-0004. https://jeffe.cs.illinois.edu/open/.
• S3 Jeff Erickson, Faces of Intricate Polytopes. https://jeffe.cs.illinois.edu/open/intricate.html.