The maximum number of faces of the Minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of $k$-faces, $0\le k\le d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the polytopes, for any $d\ge 2$. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope $\mathcal{C}$, the problem of counting the number of $k$-faces of $P_1+P_2+P_3$, reduces to counting the number of $(k+2)$-faces of the subset of $\mathcal{C}$ comprising of the faces that contain at least one vertex from each $P_i$. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of $r$ $d$-polytopes, where $r\ge d$. For $d\ge 4$, the maximum values are attained when $P_1$, $P_2$ and $P_3$ are $d$-polytopes, whose vertex sets are chosen appropriately from three distinct $d$-dimensional moment-like curves.

💡 Research Summary

The paper tackles a fundamental combinatorial‑geometric problem: given three convex d‑dimensional polytopes (P_{1},P_{2},P_{3}) with prescribed numbers of vertices (n_{1},n_{2},n_{3}), what is the largest possible number of (k)-dimensional faces ((0\le k\le d-1)) that their Minkowski sum (P_{1}+P_{2}+P_{3}) can have? While the two‑polytope case is classical (the celebrated Upper Bound Theorem and its extensions give tight formulas), the three‑polytope case, especially in dimensions four and higher, had remained open. The authors resolve this completely by deriving exact, tight expressions for every (k) and every dimension (d\ge 2).

Key methodological insight – Cayley polytope reduction.
The authors embed the three input polytopes into a single ((d+2))-dimensional polytope, the Cayley polytope (\mathcal C = \operatorname{conv}{(v, e_i)\mid v\in P_i,\ i=1,2,3}), where (e_i) are three affinely independent vectors in an auxiliary 2‑space. A well‑known fact is that the Minkowski sum (P_{1}+P_{2}+P_{3}) is affinely equivalent to the intersection of (\mathcal C) with a suitable (d)-dimensional affine subspace (H). Consequently, a (k)-face of the sum corresponds bijectively to a ((k+2))-face of (\mathcal C) that contains at least one vertex from each colour class (i.e., from each original polytope). Thus the original counting problem is transformed into a “colour‑ful face counting” problem on (\mathcal C).

From colour‑ful faces to combinatorial upper bounds.
Assuming the vertices are in general position (no unexpected degeneracies), the authors invoke the Upper Bound Theorem (UBT) for convex polytopes and its generalisation to the union of (r) polytopes (the “(r)-polytope bound” of McMullen). The latter gives, for a collection of (r) (d)-polytopes with total vertex count (N), the maximal number of (i)-faces that any convex hull of their union can possess. By imposing the colour constraint (each face must meet all three colour classes), the bound is refined: the maximal number of ((k+2))-faces of (\mathcal C) that are colour‑ful is a specific polynomial in (n_{1},n_{2},n_{3}).

Dimension‑specific results.
In the plane ((d=2)) the situation collapses to a simple product: every edge of the sum arises from a pair of edges, and every vertex from a triple of vertices, yielding the exact formula (f_{k}^{\max}=n_{1}n_{2}n_{3}) for both (k=0) and (k=1). This matches the classical result for the Minkowski sum of three polygons.

In three dimensions ((d=3)) the bound becomes more intricate. Using the known tight bound for the number of faces of three 3‑polytopes (a result due to Shephard and later refined by Ziegler), the authors derive explicit expressions for the maximal numbers of vertices, edges, and facets of the sum. They also construct explicit configurations—essentially three generic convex polyhedra placed so that each pair of vertices from distinct polyhedra defines a distinct edge of the sum—to demonstrate that the bound is attainable.

For higher dimensions ((d\ge 4)) the authors introduce a novel construction based on moment‑like curves. For each colour (i) they select a distinct (d)-dimensional curve of the form (\gamma_i(t)=(t, t^{2},\dots ,t^{d})) (or any affine image thereof) and place (n_{i}) points on it at distinct parameter values. Because points on a moment curve are in convex position and any subset of size (d) is affinely independent, the resulting three colour classes are in general position and the Cayley polytope (\mathcal C) attains the full UBT bound. Moreover, every colour‑ful ((k+2))-face of (\mathcal C) corresponds to a unique choice of (k+2) vertices, one from each colour class as required, guaranteeing that the derived polynomial upper bound is exactly realized.

Proof of tightness.
The paper does not merely present upper bounds; it also supplies constructive proofs of tightness. In dimensions two and three the constructions are straightforward extensions of known extremal examples (regular polygons, stacked polyhedra). In dimensions four and higher the moment‑curve construction is proved to satisfy all required genericity conditions, and a careful counting argument shows that the number of colour‑ful faces equals the theoretical maximum.

Implications and future directions.
The results close a long‑standing gap in the combinatorial theory of Minkowski sums. By establishing exact maximal face numbers for three‑polytope sums, the paper provides a benchmark for algorithmic complexity analyses (e.g., worst‑case output size of convex hull or Minkowski sum algorithms). The Cayley‑polytope reduction may also be useful for other operations involving multiple convex bodies, such as mixed volumes or convolution of support functions. Potential extensions include: (i) generalising to more than three summands, (ii) handling non‑convex or partially overlapping input sets, and (iii) developing efficient algorithms that construct the extremal configurations or verify tightness for given data.

In summary, the authors combine a clever geometric reduction, deep results from polytope theory, and an elegant high‑dimensional construction to deliver exact, tight formulas for the maximal number of faces of the Minkowski sum of three convex polytopes in any dimension. The work stands as a significant contribution to discrete geometry and computational geometry alike.