The Beauty of Random Polytopes Inscribed in the 2-sphere

Universality theorems for inscribed polytopes and Delaunay triangulations

We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of $\mathbb{Q}$ are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a dire…

Neighborly inscribed polytopes and Delaunay triangulations

We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that there are superexponentially many combinatorially distinct neighborly polytopes that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of poly…

内接于单位球面$\n\mathbb{S}^2$的等面多面体的表面积

Surface areas of equifacetal polytopes inscribed in the unit sphere $\mathbb{S}^2$

研究单位球面上7、8个点的凸包表面积最大化问题，给出了凸包为全等等腰或等边三角形面时的最优解。

Inscribing a regular octahedron into polytopes

We prove that any simple polytope (and some non-simple polytopes) in $\mathbb R^3$ admits an inscribed regular octahedron.

The volume of random polytopes circumscribed around a convex body

Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We achieve these results by first proving similar statements for weighted mean width approximations of convex bodies that admit a rollin…

内接于球面的9顶点最大体积多面体

The maximum volume polytope with nine vertices inscribed in the sphere

该研究解决了凸几何中9顶点内接球面最大体积多面体的经典问题，结合组合与几何方法，确定了其体积上界及对应构型。

Containment and inscribed simplices

Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <= n-1. It is also shown that the following two statements are equivalent: (i) For every polytope Q inside K having at most d+1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace W, the orthogonal projection of the set L onto W contains a translate of the corresponding projection of the set K onto W. It is then shown that, if K is a co…

Macbeath定理在有限维规范空间中的变体

Variants of a theorem of Macbeath in finite dimensional normed spaces

本文研究了在有限维规范空间中，最大体积凸多面体的几种体积指标的极值问题，扩展了Macbeath定理的范围。

一个多面体适用于所有：通过同时内在体积近似来表征欧几里得球体

Simultaneous random polytope approximation characterizes the Euclidean ball

本文通过同时内在体积近似证明欧几里得球体的唯一性，揭示了内接多面体在渐近最优条件下的刚性特性。

The Geometry of $H_4$ Polytopes

We describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In $§$7 we apply our results to the even unimodular lattice $E_8$ and show how the 600-cell transforms $E_8$/2$E_8$, an 8-space over the field $\bf{F}$$_2$, into a 4-space over $\bf{F}$$_4$ whose points, lines and planes are labeled by the geometric objects of the 600-cell.

An extremum property characterizing the n-dimensional regular cross-polytope

In the spirit of the Genetics of the Regular Figures, by L. Fejes Tóth, we prove the following theorem: If $2n$ points are selected in the $n$-dimensional Euclidean ball $B^n$ so that the smallest distance between any two of them is as large as possible, then the points are the vertices of an inscribed regular cross-polytope. This generalizes a result of R. A. Rankin for $2n$ points on the surface of the ball. We also generalize, in the same manner, a theorem of Davenport and Hajós on a set of $n+2$ points. As a corollary, we obtain a solution to the problem of packing $k$ unit $n$-dimensional…

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…

凸体内部及边界上随机点的凸函数不等式

Inequalities for convex functions of random points inside and on the boundary of convex bodies

研究凸体内部及边界上随机点，证明在特定条件下\(I\)在凸序上被\(B\)支配，得出对于凸函数\(\varphi\)的期望不等式，还得到随机弦矩及随机单纯形平均体积的不等式，证实相关猜想。

球面上的最小色散

Minimal dispersion on the sphere

研究球面上点集的色散性质，通过随机选取和补充点来分析色散上界，并探讨其与球面覆盖和多面体近似的关系。