On symmetric versions of Sylvester's problem

Sylvester-Gallai定理的一个构造性版本

A constructive version of the Sylvester-Gallai Theorem

本文从构造性数学视角审视Sylvester-Gallai定理，指出经典证明的非构造性，并利用Bishop的构造性方法建立该定理在平面上的构造性版本，揭示其隐藏的构造性内容。

Sylvester问题的细化：组合类型的概率

A refinement of the Sylvester problem: Probabilities of combinatorial types

本文研究了Sylvester问题的细化版本，计算了随机点的凸包具有特定组合类型的概率，考虑了正态分布、Beta分布和随机游走等不同情况。

A centrally symmetric version of the cyclic polytope

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 0< j <k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least (c_j(d)+o(1)) {n \choose j+1} for some c_j(d)>0 and at most (1-2^{-d}+o(1)){n \choose j+1}…

On a conjectural symmetric version of Ehrhard's inequality

We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting $J_{k-1}(s)=\int^s_0 t^{k-1} e^{-\frac{t^2}{2}}dt$ and $c_{k-1}=J_{k-1}(+\infty)$, we conjecture that the function $F:[0,1]\rightarrow\mathbb{R},$ given by $$F(a)= \sum_{k=1}^n 1_{a\in E_k}\cdot(β_k J_{k-1}^{-1}(c_{k-1} a)+α_k)$$ (with an appropriate choice of a decomposition $[0,1]=\cup_{i} E_i$ and coefficients $α_i, β_i$) satisfies, for all symmetric convex sets $K$ and $L,$ and any $λ\in[0,1]$, $$ F\left(γ(λK+(1-λ)L)\right)\geq λF\left(…

Symmetric versions of Laman's Theorem

Recent work has shown that if an isostatic bar and joint framework possesses non-trivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are `fixed' by various symmetry operations of the framework. For the group $C_3$ which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed in [4] that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with $C_3$ symmetry to be isostatic, provided that its joints are positioned as ge…

An isomorphic version of the slicing problem

Here we show that any n-dimensional centrally symmetric convex body K has an n-dimensional perturbation T which is convex and centrally symmetric, such that the isotropic constant of T is universally bounded. T is close to K in the sense that the Banach-Mazur distance between T and K is O(log n). If K has a non-trivial type then the distance is universally bounded. In addition, if K is quasi-convex then there exists a quasi-convex T with a universally bounded isotropic constant and with a universally bounded distance to K.