Face Bounds for Minkowski Sums of Polytopes

Definitions and mathematical statement

Fix positive integers $d,n$. Let $P_1,\ldots,P_n\subset{\mathbb{R}}^d$ be convex $d$-polytopes, so each is full-dimensional, with $v_i=f_0(P_i)$ vertices. Their Minkowski sum is \[ P=P_1+\cdots+P_n =\{x_1+\cdots+x_n:x_i\in P_i\}. \] For $0\leq k\leq d-1$, let $f_k(P)$ be the number of distinct $k$-dimensional faces of $P$. The requested optimal bound can be expressed, for every feasible vertex-count tuple, as \[ B_{d,n,k}(v_1,\ldots,v_n) =\max\left\{f_k(Q_1+\cdots+Q_n): Q_i\subset{\mathbb{R}}^d\text{ convex }d\text{-polytope},\ f_0(Q_i)=v_i\right\}. \] The problem is to determine these sharp upper bounds in terms of the individual vertex numbers, together with constructions proving sharpness in the intended sense. Feasible means that full-dimensional polytopes with the specified counts exist; no assertion is made for an empty parameter class.

The range includes every face dimension below $d$ and arbitrary numbers of summands. It is not restricted to two summands, $n\leq d$, simplicial/simple polytopes or a general-position subclass. Faces are counted in the sum itself, not by potentially repeated decompositions into summand faces. A formula sharp separately for each $k$ and a single configuration simultaneously attaining all bounds are different requirements; the source does not explicitly demand the latter or clarify exact versus asymptotic sharpness, so no additional simultaneous-extremizer condition is inserted.

The supplied literature triage checked 21 August 2026 labels the extracted question open and reports no authoritative complete resolution found. It cites only the historical report and supplies no separate bound theorem or case list. That dated claim is retained as supplied metadata rather than an independently verified current literature assessment.

Short English statement

Determine the largest possible number of faces in each dimension of a Minkowski sum, knowing the number of vertices of each full-dimensional summand. The bound must cover every number of summands and the full face range; the supplied open label is a dated assessment requiring independent verification.

Sources

• Local UnsolvedMath registry, permanent ID 30000684, exact source code catalogue entry; Face Bounds for Minkowski Sums of Polytopes. Proposed year 2007, status open; corrected_verified extraction retains all summands, vertex counts and $0\leq k\leq d-1$.
• Geometric and Topological Combinatorics (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/04, pp. 195--272, https://doi.org/10.4171/owr/2007/04.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2007/04, and reports no authoritative complete resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no separate modern theorem citation is supplied.