The 4-D Gaussian Random Vector Maximum Conjecture and the 3-D Simplex Mean Width Conjecture

Mean Width of a Regular Simplex

The mean width is a measure on n-dimensional convex bodies. An integral formula for the mean width of a regular n-simplex appeared in the electrical engineering literature in 1997. As a consequence, expressions for the expected range of a sample of n+1 normally distributed variables, for n<=6, carry over to widths of regular n-simplices. As another consequence, precise asymptotics for the mean width become available as n->infty.

The Junta Method for Hypergraphs and the Erdős-Chvátal Simplex Conjecture

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a $k$-uniform hypergraph on $n$ vertices that does not contain an `enlarged' copy $H^+$ of a fixed hypergraph $H$. These include well-known problems such as the Erdős-Sós `forbidding one intersection' problem and the Frankl-Füredi `special simplex' problem. We present a general approach to such problems, using a `junta approximation method' that originates from analysis of Boolean functions. We prove that any $H^+$-free hypergraph is essentially contained in a `junta' -- a hypergraph determined by a small numb…

高斯最大值的随机控制：对弱单纯形猜想的解决

Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture

研究高斯最大值的随机比较，通过结合高斯乘积不等式与自适应倾斜论证，解决了弱单纯形猜想，证明了单纯形平均宽度猜想中的不等式，并给出特定条件下可发送等概率消息最大数量的精确公式。

关于随机单形期望体积的单调性

On the monotonicity of the expected volume of a random simplex

设d维凸体中的随机单形是该凸体上d+1个随机点的凸包。研究了随机单形的期望体积作为凸体的函数在包含下是否单调非减的问题。我们证明，如果d是1或2，这成立，如果d >= 4，则不成立。我们还证明了类似的结果，为更高的时刻的体积的随机单形，特别是二阶矩，这对应于行列式的协方差阵的凸体。这些问题的动机是切片猜想。

Mean width of regular polytopes and expected maxima of correlated Gaussian variables

An old conjecture states that among all simplices inscribed in the unit sphere the regular one has the maximal mean width. An equivalent formulation is that for any centered Gaussian vector $(ξ_1,\dots,ξ_n)$ satisfying $\mathbb Eξ_1^2= \dots =\mathbb Eξ_n^2=1$ one has $$ \mathbb E\,\max\{ξ_1,\dots,ξ_n\}\leq\sqrt{\frac{n}{n-1}}\, \mathbb E\,\max\{η_1,\dots,η_n\}, $$ where $η_1,η_2,\dots,$ are independent standard Gaussian variables. Using this probabilistic interpretation we derive an asymptotic version of the conjecture. We also show that the mean width of the regular simplex with $2n$ vertice…

Mean Width of a Regular Cross-Polytope

The expected range of a sample of n+1 normally distributed variables is known to be related to the mean width of a regular n-simplex. We show that the expected maximum mu_n of a sample of n half-normally distributed variables is related to the mean width of a regular n-crosspolytope. Both of these relations have mean square counterparts. An expression for mu_5 is found and is believed to be new.