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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 . 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 . Our main tool is the observation that the Gale dual of independent samples in itself can be taken to be a set of independent points (translated to have barycenter at the origin) when the distribution of the points is Gaussian.more » « less
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We show that any embedding $$\mathbb {R}^d \to \mathbb {R}^{2d+2^{\gamma (d)}-1}$$ inscribes a trapezoid or maps three points to a line, where $$2^{\gamma (d)}$$ is the smallest power of $$2$$ satisfying $$2^{\gamma (d)} \geq \rho (d)$$ , and $$\rho (d)$$ denotes the Hurwitz–Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely $$3$$ -regular maps, for infinitely many dimensions $$d$$ , without resorting to sophisticated algebraic techniques.more » « less
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Abstract We provide a simple characterization of simplicial complexes on few vertices that embed into thed-sphere. Namely, a simplicial complex on$$d+3$$ vertices embeds into thed-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen–Flores theorem and provide a topological extension of the Erdős–Ko–Rado theorem. By analogy with Fáry’s theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.more » « less
