Convexly independent subsets of Minkowski sums of convex polygons

We show that there exist convex $n$-gons $P$ and $Q$ such that the largest convex polygon in the Minkowski sum $P+Q$ has size $\Theta(n\log n)$. This matches an upper bound of Tiwary.

💡 Research Summary

The paper investigates the maximal size of a convexly independent subset (denoted ci(X)) inside the Minkowski sum of two convex polygons. For a finite planar point set X, ci(X) is the cardinality of the largest subset that forms a convex polygon. Classical results such as the Happy Ending theorem guarantee that ci(X) grows without bound as |X| increases, with a logarithmic lower bound in the worst case. When X is a Minkowski sum P+Q of two point sets of size n, previous work by Eisenbrand et al. showed an upper bound O(n^{4/3}) on ci(P+Q), and a matching construction was later given by Bílka et al. However, those constructions allowed one of the summands to be non‑convex. The natural open question is: what is the largest possible ci(P+Q) when both P and Q are convex n‑gons?

Tiwary (2014) proved an upper bound ci(P+Q)=O((n+m)·log(n+m)) for a convex n‑gon P and a convex m‑gon Q, but left open whether this bound is tight. The present work resolves this by constructing explicit families of convex polygons that achieve Θ(n log n), thereby showing Tiwary’s bound is optimal.

The core of the construction relies on a geometric object called a “south‑east chain”. A south‑east chain is a sequence of points (a^{(1)},…,a^{(n)}) in the plane such that both coordinates are strictly increasing and the slopes of consecutive segments form a strictly increasing sequence. Lemma 2.2 establishes that any south‑east chain is convexly independent: its convex hull has exactly n vertices.

The authors then introduce three elementary transformations that preserve the south‑east‑chain property while dramatically altering slopes:

1. Flattening: L_ε(x,y) = (εx, ε²y) with ε>0 makes the chain arbitrarily flat (slopes tend to 0) while preserving the order.
2. Rotation: R is the counter‑clockwise rotation by 60° about the origin. When applied to a sufficiently flat chain, the rotated chain remains a south‑east chain, but its slopes become close to √3.
3. Midpoint combination: Given a chain A and its rotated copy A′, the sequence (a_i + a′_i)/2 forms another south‑east chain whose slopes converge to 1/√3.

Lemma 2.3 provides the trigonometric justification: if a segment has slope tan θ with θ<π/6, then after rotation its slope becomes tan(π/3+θ), and the slope of the midpoint segment becomes tan(π/6+θ). Lemma 2.4 formalizes that for sufficiently small ε, the three transformed sequences (A_ε, A′_ε, A′′_ε) are all south‑east chains with limiting slopes 0, √3, and 1/√3 respectively.

Using these tools, the authors construct three chains A, B, C with |A|=|B|=n and C⊂(A+B)/2. After flattening and rotating A and B, they translate the rotated copies by fixed vectors (0,2) and (1,1) respectively, and concatenate the original and translated chains to obtain new chains à and B̃. Lemma 2.5 shows that à and B̃ are still south‑east chains and that (Ã+B̃)/2 contains a south‑east chain of length at least 2|C|+|A|. In other words, the length of the middle chain roughly doubles while the outer chains double in size.

Iterating this construction k times yields convex point sets P_k and Q_k with |P_k|=|Q_k|=2^k and a convexly independent subset of P_k+Q_k of size (k+2)·2^k−1. Substituting n=2^k gives ci(P_k+Q_k)=Θ(n log n). Thus the authors exhibit convex n‑gons whose Minkowski sum contains a convex polygon with Θ(n log n) vertices, matching Tiwary’s upper bound.

The paper also interprets the construction in graph‑theoretic terms. Consider a bipartite graph G=(U∪V,E) where U and V correspond to the two chains and an edge (u,v) exists iff the midpoint (u+v)/2 belongs to the middle chain. The construction yields graphs with |U|=|V|=n and |E|≈n log n, while all vertices and all midpoints lie in convex position. This contrasts with the notion of a strong convex embedding, where the edge count is linear; the present “relaxed” embedding permits a logarithmic factor.

Finally, the authors discuss connections to the classic unit‑distance problem of Erdős and Moser. Since ci(P+(−P)) bounds the number of unit‑distance pairs in a convex n‑gon, one might hope to improve the known O(n log n) bound using their construction. However, they observe that their graphs do not directly yield unit‑distance configurations, leaving the problem open.

In summary, the paper provides a clean geometric construction that proves the Θ(n log n) lower bound for convexly independent subsets of Minkowski sums of convex polygons, thereby confirming the optimality of the previously known upper bound. The work blends elementary linear transformations, careful slope analysis, and iterative concatenation to achieve the result, and it also sheds light on related topics such as convex embeddings of graphs and the unit‑distance problem.