On sequences of convex records in the plane

TL;DR

This work studies convex records in the plane, where a point is a convex record if it lies outside the convex hull of all preceding points. Using an exact identity that links the mean number of hull vertices $\langle N_n\rangle$ to the mean number of convex records $\langle R_n\rangle$ for iid data, the authors perform a comprehensive numerical and analytical survey across four iid planar distributions (square, disk, Gaussian, isotropic power-law) and analyze planar random walks. They derive and validate growth laws for $\langle N_n\rangle$ and $\langle R_n\rangle$, show finite, distinct Fano factors, and reveal universal limit distributions for the ratio $R_n/\langle R_n\rangle$ in random walks, along with detailed extremal probabilities for extreme hull configurations. The results highlight affine invariance, connect convex hull geometry with multivariate record statistics, and provide a framework for higher-dimensional generalizations and rigorous proofs. Overall, the paper unifies geometric extremes and records in the plane and exposes universal statistical structures across diverse sampling schemes.

Abstract

Convex records have an appealing purely geometric definition. In a sequence of $d$-dimensional data points, the $n$-th point is a convex record if it lies outside the convex hull of all preceding points. We specifically focus on the bivariate (i.e., two-dimensional) setting. For iid (independent and identically distributed) points, we establish an identity relating the mean number $\mean{R_n}$ of convex records up to time $n$ to the mean number $\mean{N_n}$ of vertices in the convex hull of the first $n$ points. By combining this identity with extensive numerical simulations, we provide a comprehensive overview of the statistics of convex records for various examples of iid data points in the plane: uniform points in the square and in the disk, Gaussian points and points with an isotropic power-law distribution. In all these cases, the mean values and variances of $N_n$ and $R_n$ grow proportionally to each other, resulting in finite limit Fano factors $F_N$ and $F_R$. We also consider planar random walks, i.e., sequences of points with iid increments. For both the Pearson walk in the continuum and the Pólya walk on a lattice, we characterise the growth of the mean number $\mean{R_n}$ of convex records and demonstrate that the ratio $R_n/\mean{R_n}$ keeps fluctuating with a universal limit distribution.

Figures (19)

• Figure 1: The three possible cases of convex records for $n=4$ (see (\ref{['nr4']})). The fourth data point and the attached edges of the convex hull, if any, are shown in colour. Left to right: $(N_4=3,\,R_4=3)$, $(N_4=3,\,R_4=4)$, $(N_4=4,\,R_4=4)$.
• Figure 2: The 36 possible values taken by the couple $(N_n,R_n)$ for $n=10$.
• Figure 3: A sample of 200 uniform iid points in the unit square such that $N=11$ and $R=38$. Green polygon: convex hull of the dataset. Green symbols: the 11 vertices of the convex hull. Red symbols: the 27 other convex records. Blue symbols: the 162 data points that are not convex records.
• Figure 4: Numbers $N_n$ (lower tracks) and $R_n$ (upper tracks) plotted against time $n$ for three histories of 1,000 uniform data points in the unit square. Each history is shown by a colour.
• Figure 5: ${\langle N_n\rangle}$ and ${\mathop{\rm var}\,}{N_n}$ plotted against $\ln n$ for uniform points in the unit square. Dashed lines show linear fits, slightly offset for greater clarity.