[Paper Review] Monotonicity of the Sample Range of 3-D Data: Moments of Volumes of Random Tetrahedra

Stefan Kunis, Benjamin Reichenwallner+1 more|arXiv (Cornell University)|Dec 6, 2016

Point processes and geometric inequalities6 references3 citations

TL;DR

This paper disproves the conjecture that the expected volume of the convex hull of random points in a convex body is monotone with respect to set inclusion in three dimensions. Using an infinitesimal perturbation of a tetrahedron and computing even moments of the volume of random simplices with one vertex fixed at a facet centroid, the authors construct a counterexample where the expected volume in a smaller set exceeds that in a larger one, resolving an open question posed by Meckes and providing an explicit formula for even moments of such volumes in the tetrahedron.

ABSTRACT

The sample range of uniform random points $X_1, \dots , X_n$ chosen in a given convex set is the convex hull ${ m conv}[X_1, \dots, X_n]$. It is shown that in dimension three the expected volume of the sample range is not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is the three-dimensional tetrahedron together with an infinitesimal variation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.

Motivation & Objective

• To resolve whether the expected volume of the convex hull of i.i.d. uniform random points in a convex body is monotone under set inclusion in three dimensions.
• To construct a counterexample where a smaller convex body has a larger expected convex hull volume than a larger one, thereby refuting a conjecture by Meckes.
• To compute the even moments of the volume of a random simplex formed by three uniform random points in a tetrahedron and the centroid of one of its facets.
• To develop and apply a polynomial approximation technique to bound the expected volume from above using moment information.
• To provide an explicit formula for all even moments of the volume of such a random simplex in a tetrahedron, enabling precise numerical verification of the counterexample.

Proposed method

• Utilizes Rademacher's lemma, which reduces the monotonicity question to verifying whether the expected volume of a random simplex with one vertex fixed at a boundary point is bounded above by the expected volume with that point replaced by a random point.
• Fixes one vertex of the random simplex at the centroid of a facet of the tetrahedron, exploiting symmetry to simplify computation.
• Computes the even moments of the volume of the random simplex with one fixed vertex using analytical integration techniques and symbolic computation.
• Applies a one-sided polynomial approximation to the absolute value function on $[-1/3, 1/3]$, using Hermite interpolation to ensure $P(x) o |x|$ from above, enabling upper bounds on the expected volume.
• Solves a finite-dimensional linear program to find the optimal polynomial approximation with $n=13$ and $L=1000$ interpolation nodes, rationalizing the nodes to obtain a polynomial with rational coefficients.
• Uses the computed moments and the polynomial bound to show that $\mathbb{E}|\text{conv}[X_1,X_2,X_3,c]| < \mathbb{E}|\text{conv}[X_1,X_2,X_3,X_4]|$, thus proving the counterexample.

Experimental results

Research questions

• RQ1 Is the expected volume of the convex hull of $n=4$ i.i.d. uniform random points in a convex body $K \subset \mathbb{R}^3$ monotone with respect to set inclusion?
• RQ2 Can a counterexample be constructed where a smaller convex body $L \subset K$ has a larger expected convex hull volume than $K$?
• RQ3 What are the even moments of the volume of a random simplex formed by three uniform random points in a tetrahedron and the centroid of one of its facets?
• RQ4 Can polynomial approximation techniques be used to bound the expected volume of such a simplex from above using only moment information?
• RQ5 Does the explicit formula for the even moments of the volume in the tetrahedron setting allow for a numerical verification of non-monotonicity?

Key findings

• The expected volume of the convex hull of four uniform random points in a three-dimensional tetrahedron is $\frac{13}{720} - \frac{\pi^2}{15015} \approx 0.01739$.
• The expected volume of a random simplex with three uniform random points in a tetrahedron and the centroid of one facet is bounded above by $0.0173791\ldots$ using a degree-26 even polynomial approximation.
• This upper bound is strictly less than the expected volume in the full tetrahedron, proving that $\mathbb{E}|\text{conv}[X_1,X_2,X_3,c]| < \mathbb{E}|\text{conv}[X_1,X_2,X_3,X_4]|$.
• The counterexample is constructed by considering an infinitesimal perturbation of the tetrahedron, showing that the expected volume is not monotone under inclusion in $\mathbb{R}^3$.
• An explicit formula is derived for all even moments of the volume of the random simplex with one vertex fixed at a facet centroid, enabling precise numerical verification.
• The method successfully overcomes the intractability of direct volume computation by combining moment computation with polynomial approximation, providing a rigorous bound.