A Carousel Property for Compact Convex Sets

TL;DR

The authors generalize the weak carousel rule from disks in a triangle to arbitrary compact convex sets A_0,A_1 contained in a convex n-gon G, showing that if the number s of common supporting lines is less than n, then there exist i∈{0,1} and j∈{1,...,n} with A_i ⊆ Conv(A_{1−i}, Vert(G)\{g_j}). They develop a geometric framework based on support functions, common supporting lines, sectors, and slide-turning to control how sectors can expand and intersect G’s boundary, enabling a combinatorial-geometric proof that yields the desired containment. The main result subsumes and extends Adaricheva–Bolat and Czédli’s work, provides sharpness for even n, and yields corollaries for algebraic-curve boundaries and ellipses; the paper also raises several open questions about odd n, higher-dimensional analogues, and convex-hull generalizations. Overall, the work advances understanding of how complex convex shapes interact inside polygonal envelopes and informs representation questions for convex geometries.

Abstract

We prove that if $A_0$ and $A_1$ are compact convex sets contained in a convex $n$-gon with vertices $g_1, \dots, g_n$, and $n$ is strictly greater than the number of common supporting lines of $A_0$ and $A_1$, then there exist $i \in \{0,1\}$ and $j \in \{1,\dots, n\}$ such that $A_i$ is in the convex hull of $A_{1-i}$ and $(\{g_1, \dots, g_n\} \setminus \{g_j\})$. This recovers and generalizes previous results of Adaricheva--Bolat and Cz{é}dli. We also show that this bound is sharp for even $n$.

Figures (7)

• Figure 1: On the left, $l_i$ and $l_{i+1}$ are adjacent, for $i=1,2,3,4$, indices $\ (\mathrm{mod}\ 4)$. On the right, the sector $\mathop{\mathrm{Sect}}\nolimits^+(l_1,l_2; B)$ extends along $\mathop{\mathrm{dir}}\nolimits^L(l_1)$ and $\mathop{\mathrm{dir}}\nolimits^L(l_2)$.
• Figure 2: $\mathop{\mathrm{Sect}}\nolimits^+(l_1,l_2; A_1)\cap G$ in dark gray, and the expanded $\mathop{\mathrm{Sect}}\nolimits^+(l_{1,\alpha; A_1}, l_{2,-\beta; A_1}; A_1) \cap G$ in light gray.
• Figure 3: The vertices between $l_1$ and $l_2$ are dotted, and the sectors $\mathop{\mathrm{Sect}}\nolimits^+(l_1, l_2; A_1)$ are shaded. Note the difference in order depending on whether the sector intersects $G$.
• Figure 4: On the left, $\mathop{\mathrm{sweep}}\nolimits^L(l_1)$ in blue and $\mathop{\mathrm{sweep}}\nolimits^L(l_2)$ in red. On the right, $\mathop{\mathrm{sweep}}\nolimits^R(l_1)$ in blue and $\mathop{\mathrm{sweep}}\nolimits^R(l_4)$ in red.
• Figure 5: The ellipses $A_0, A_1$ shaded, the ellipses $X_1, X_2, X_3$ unshaded. The weak carousel rule fails for $\mathcal{A} = \{A_0, A_1\}$ and $G = \mathop{\mathrm{Conv}}\nolimits(X_1, X_2, X_3)$.

Theorems & Definitions (29)

• Theorem : Theorem 3.1, ADARICHEVA2019726
• Theorem 1.1
• Theorem 2.1: Rockafellar1970
• Lemma 2.1: tropp, § 2.5
• Corollary 2.1
• Lemma 2.2
• proof
• Proposition 2.1
• proof
• Theorem 2.2