Beta Polytopes and Beta Cones: An Exactly Solvable Model in Geometric Probability

TL;DR

The paper provides an exact, nonasymptotic theory for random beta polytopes generated by points with beta distributions in the unit ball, and introduces beta cones as the key analytic bridge to conic geometry. Central to the approach is the Theta-function, which captures a wide class of angular and conic-intrinsic-volume functionals, allowing explicit expressions for quantities such as f_k, Vol_d, V_ℓ, angle sums, conic intrinsic volumes, Beta content, and the S-functional. The authors develop a comprehensive framework linking polytopes and cones via canonical Ruben–Miles decomposition, projection identities, and general-position results, yielding precise formulas even in mixed-parameter, nonidentical-beta settings. They also analyze two extremal regimes (β = -1 and β → ∞) and solve a Sylvester-type simplex problem in this mixed-beta setting, illustrating the breadth and solvability of the model. The methods fuse stochastic geometry, integral geometry, and special-function analysis, producing a versatile exact solvable model with potential connections to Gaussian polytopes and higher-dimensional projection phenomena.

Abstract

Let $X_1,\ldots, X_n$ be independent random points in the unit ball of $\mathbb R^d$ such that $X_i$ follows a beta distribution with the density proportional to $(1-\|x\|^2)^{β_i}1_{\{\|x\| <1\}}$. Here, $β_1,\ldots, β_n> -1$ are parameters. We study random polytopes of the form $[X_1,\ldots,X_n]$, called beta polytopes. We determine explicitly expected values of several functionals of these polytopes including the number of $k$-dimensional faces, the volume, the intrinsic volumes, the total $k$-volume of the $k$-skeleton, various angle sums, and the $S$-functional which generalizes and unifies many of the above examples. We identify and study the central object needed to analyze beta polytopes: beta cones. For these, we determine explicitly expected values of several functionals including the solid angle, conic intrinsic volumes and the number of $k$-dimensional faces. We identify expected conic intrinsic volumes of beta cones as a crucial quantity needed to express all the functionals mentioned above. We obtain a formula for these expected conic intrinsic volumes in terms of a function $Θ$ for which we provide an explicit integral representation. The proofs combine methods from integral and stochastic geometry with the study of the analytic properties of the function $Θ$.

Figures (5)

• Figure 1.1: A possible realization of $\mathscr{C} \sim \operatorname{BetaCone}(\mathbb{R}^3; \beta;\beta_1,\dots,\beta_6)$. Actually, the figure shows $\mathscr{C}+Z$, a beta cone shifted by the vector $Z$. The apex of the original beta cone is at $0$.
• Figure 1.2: The expected volume of $\mathscr{P}_{12,2}^{\beta_1,\dots,\beta_{12}}$ divided by the area of the unit disk is the probability that the additional point $Y$ takes its value inside of $\mathscr{P}_{12,2}^{\beta_1,\dots,\beta_{12}}$.
• Figure 1.3: Illustration of Example \ref{['ex:expect_vol_as_angle']} for $d=1$ and $d=2$.
• Figure 3.1: Visualization of the construction used in the proof of Theorem \ref{['theo:representation_angles_and_construction_of_Cone']}. Here, the gray cone is $\mathscr{C}=\mathop{\mathrm{pos}}\nolimits(Z_1-Z,\ldots, Z_5-Z) \sim \operatorname{BetaCone}(\mathbb{R}^3; \beta;\beta_1,\dots,\beta_5)$ and the continuous red line is the face $G=\mathop{\mathrm{pos}}\nolimits (Z_1-Z)$. Its continuation is the affine subspace $A=\mathop{\mathrm{aff}}\nolimits (Z,Z_1)$, which is also the lineality space of the tangent cone $T(G,\mathscr{C})$. The green plane is the orthogonal complement $A^\perp$ of the line $A$. We project each point $Z,Z_1,\dots,Z_5$ onto $A^\perp$ which is identified with $\mathbb{R}^2$. The projections $Y,Y_2,\dots,Y_5$ are beta-distributed as stated in Proposition \ref{['prop:beta_cones_joint_distr_projections']}. Thus, the isometry of $T(G,\mathscr{C})$ to $\mathbb{R} \oplus \operatorname{BetaCone}(\mathbb{R}^2; \beta+\beta_1+2;\beta_2+\frac{1}{2},\dots,\beta_5+\frac{1}{2})$ becomes clear.
• Figure 6.1: Visualization of the proof of Theorem \ref{['theo:tangent_cone_is_beta_cone']}. Here, we have a beta polytope $\mathscr{P}=\mathscr{P}_{9,3}^{\beta_1,\dots,\beta_9}$. The simplex $F=[X_1,X_2,X_3]$ is a 2-dimensional face of the polytope. The red plane is the affine subspace $A=\mathop{\mathrm{aff}}\nolimits(X_1,X_2,X_3)$, and its orthogonal complement is the green line. The projected points $Y_4, \dots, Y_9$ are the orange dots on the dashed green line. The cone $\mathop{\mathrm{pos}}\nolimits(Y_4-Y,\dots,Y_9-Y)$ is the part of the green line that is dashdotted. One can clearly see that the blue tangent cone $T(F,\mathscr{P})$ (which is in fact a half-space) is isometric to $(A-\bar{X}) \oplus \mathop{\mathrm{pos}}\nolimits(Y_4-Y,\dots,Y_9-Y)$, where $\bar{X}=\frac{X_1+X_2+X_3}{3}$.

Theorems & Definitions (160)

• Theorem 1.1: Selected results on beta cones
• Theorem 1.2: Selected results on beta polytopes
• Remark 1.3: On beta type distributions
• Remark 1.4: On variances and moments
• Example 1.5: Expected volume as absorbtion probability
• Example 1.6: Expected volume as expected angle
• Lemma 2.1
• proof
• Corollary 2.2
• Lemma 2.3: Projection property