Interior angle sums of geodesic triangles in $S^2 \times R$ and $H^2 \times R$ geometries

cevian simplex 的体积最大值及其部分

The Maximum of the Volume of a Cevian Simplex and its Parts

研究cevian simplex的体积最大值及其部分的优化问题，利用重心坐标推广到n维简单形，最大值为1/n^n。

非零边de Sitter三角形的Girard型定理

Girard Type Theorems for de Sitter Triangles with non-null Edges

Girard定理是球面三角形内角与面积有关的定理.在本文中，我们引入了它的类似物的正常deSitter三角形的非零边。

A Mattila-Sjölin theorem for triangles

We show for a compact set $E \subset \mathbb{R}^d$, $d \geq 4$, that if the Hausdorff dimension of $E$ is larger than $\frac{2}{3}d+1$, then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. Here we understand the set of congruence classes of triangles formed by triples of points of $E$ as the set $$Δ_{\text{tri}}(E) = \left \{ (t,r, α) : |x-z|=t, |y-z|=r \, \text{ and }\, α= α(x,z,y), \ x,y,z \in E \right \},$$ where $α(x,z,y)$ denotes the angle formed by $x$, $y$ and $z$ , centered at $z$. This extends the Mattila-Sjölin theorem that establi…

某些和的内部与极薄 Cantor 集的连续像

Interior of certain sums and continuous images of very thin Cantor sets

本文通过构造极薄 Cantor 集并利用非退化 $C^1$ 映射，证明相关和集与连续像具有非空内部，进而推出拓扑 Erdős 相似性猜想的更强结论，并构造距离集含内部的 Cantor 集。

Subdividing triangles with $π$-commensurable angles

A point in the interior of a planar triangle determines a subdivision into six subtriangles. A triangle with angles commensurable with $π$ is called $π$-commensurable. For such a triangle a subdivision where each of the subtriangles are $π$-commensurable too is called $π$-commensurable. We prove that there are infinitely many $π$-commensurable triangles that do not admit any $π$-commensurable subdivision except the one given by angle bisectors. We count the number of $π$-commensurable subdivisions of triangles. We perform a similar count for Z-degree sub-divisions of Z-degree triangles too. Fi…

Maximal Angle Flow on the Shape Sphere of Triangles

There has been recent work using Shape Theory to answer the longstanding and conceptually interesting problem of what is the probability that a triangle is obtuse. This is resolved by three kissing cap-circles of rightness being realized on the shape sphere; integrating up the interiors of these caps readily yields the answer to be 3/4. We now generalize this approach by viewing rightness as a particular value of maximal angle, and then covering the shape sphere with the maximal angle flow. Therein, we discover that the kissing cap-circles of rightness constitute a separatrix. The two qualitat…

Reptilings and space-filling curves for acute triangles

An $r$-gentiling is a dissection of a shape into $r \geq 2$ parts which are all similar to the original shape. An $r$-reptiling is an $r$-gentiling of which all parts are mutually congruent. By applying gentilings recursively, together with a rule that defines an order on the parts, one may obtain an order in which to traverse all points within the original shape. We say such a traversal is a face-continuous space-filling curve if, at any level of recursion, the interior of the union of any set of consecutive parts is connected---that is, consecutive parts must always meet along an edge. Most …

Angle sums of simplicial polytopes

The interior angle vector ($\widehatα$-vector) of a polytope is a metric analogue of the $f$-vector in which faces are weighted by their solid angle. For simplicial polytopes, Dehn-Sommerville-type relations on the $\widehatα$-vector were introduced by Sommerville (1927) and Höhn (1953). Camenga (2006) defined the $\widehatγ$-vector, a linear transformation analogous to the $h$-vector and conjectured it to be non-negative. Using tools from geometric and algebraic combinatorics, we prove this conjecture and show that the $\widehatγ$-vector increases in the first half and is flawless. In contras…

Marion定理的推广：通过三等分单纯形边得到的中心多胞体的体积

Generalization of Marion's Theorem: Volumes of Central Polytopes Obtained by Trisecting the Edges of Simplices

将三角形中Marion定理推广到n维单纯形，通过三等分每条边并连接内分点得到中心多胞体，证明其体积为原单纯形体积的1/C(n,2n+1)，证明使用指数随机变量的概率解释并简化为组合二项式和。