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Youden's Demon is Sylvester's Problem
Probability · Submitted 2 Jul 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line ("Youden's demon problem") is the same as a natural generalization of Sylvester's Four Point problem to Gaussian points in $\mathbb{R}^d$.
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The abstract, sentence by sentence
If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are $\frac 6 \pi \arcsin\left(\frac{1}{3}\right)\approx .649$. We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line ("Youden's demon problem") is the same as a natural generalization of Sylvester's Four Point problem to Gaussian points in $\mathbb{R}^d$. Our main tool is the observation that the Gale dual of independent samples in $\mathbb{R}^d$ itself can be taken to be a set of independent points (conditioned on barycenter at the origin) when the distribution of the points is Gaussian.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are $\frac 6 \pi \arcsin\left(\frac{1}{3}\right)\approx .649$. | result 0.95 | 0.58 | 0.20 | asserted 1.00 |
| s1 | We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line ("Youden's demon problem") is the same as a natural generalization of Sylvester's Four Point problem to Gaussian points in $\mathbb{R}^d$. | contribution 0.59 | 0.88 | 0.83 | asserted 1.00 |
| s2 | Our main tool is the observation that the Gale dual of independent samples in $\mathbb{R}^d$ itself can be taken to be a set of independent points (conditioned on barycenter at the origin) when the distribution of the points is Gaussian. | method 0.96 | 0.81 | 0.15 | asserted 0.99 |
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Original abstract
If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are $\frac 6 \pi \arcsin\left(\frac{1}{3}\right)\approx .649$. We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line ("Youden's demon problem") is the same as a natural generalization of Sylvester's Four Point problem to Gaussian points in $\mathbb{R}^d$. Our main tool is the observation that the Gale dual of independent samples in $\mathbb{R}^d$ itself can be taken to be a set of independent points (conditioned on barycenter at the origin) when the distribution of the points is Gaussian.
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This problem was solved in the planar setting by Maehara [ref] , and later on by Blatter [ref] , who each used different methods to obtain the answer [equation] In the recent preprint [ref] , Frick, Newman, and Pegden were able to solve this Gaussian formulation of Sylvester's problem for $d=3$ by connecting it to Youden's demon problem by way of Gale duality.
Sylvester's problem for random walks and bridges · Introduction · 0.89
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