Han Duong
We extend the results of Bey, Hen, and Wills (http://arxiv.org/abs/math/0606089). In this paper, we show that, up to equivalence under unimodular transformations, there is exactly one class of $d$-simplices having $k \ge 1$ interior lattice points and minimal volume $\frac{1}{d!}(dk+1)$.
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For an integral convex polytope $\Pc \subset \RR^N$ of dimension $d$, we call $δ(\Pc)=(δ_0, δ_1,..., δ_d)$ the $δ$-vector of $\Pc$ and $\vol(\Pc)=\sum_{i=0}^dδ_i$ its normalized volume. In this paper, we will establish the new equalities and inequalities on $δ$-vectors for integral simplices whose normalized volumes are prime. Moreover, by using those, we will classify all the possible $δ$-vectors of integral simplices with normalized volume 5 and 7.
In this work, we exploit Strichartz average approach \cite{Strichartz2001} to define the Laplacian on Sierpiński gasket, in the construction of the finite volume method. The approach present sum similarities with the finite difference approach in terms of stability and convergence.
Unimodality for IDP Lattice Simplices of Prime Normalized Volume
本文证明了具有整数分解性质且正规体积为素数的格单纯形的$h^*$-多项式是单峰的，并给出了若干充分条件，回应了Ferroni的反例。研究问题源于Ehrhart理论中的猜想，核心方法是构造性证明与条件分析，主要贡献是解决了该猜想在素数体积情形下的成立性。
This note provides a simple proof for the equality between the normalized volume of a convex polytope with $m$ vertices and the mixed volume of $m$ simplices and thus shows the seemingly restrictive problem of computing mixed volume of simplices is still at least as hard as computing volumes of convex polytopes.
Sharp volume and multiplicity bounds for Fano simplices
该研究给出依赖维度和Gorenstein指数的法诺单纯形体积、马勒体积与重数的精确上界，提出高效分类方法并完成多种Gorenstein指数下四维及以下法诺单纯形的分类。
It is known that a lattice simplex of dimension $d$ corresponds a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$. Conversely, given a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$ such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension $d$. In this paper, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equals $p,p^2$ and $pq$, where $p$ and $q$ are prime numbers …
In this short note we give a construction of an infinite series of Delone simplices whose relative volume grows super-exponentially with their dimension. This dramatically improves the previous best lower bound, which was linear.
In this note we investigate the behavior of the volume that the convex hull of two congruent and intersecting simplices in Euclidean $n$-space can have. We prove some useful equalities and inequalities on this volume. For the regular simplex we determine the maximal possible volume for the case when the two simplices are related to each other via reflection at a hyperplane intersecting them.
Local $h^*$-polynomials for one-row Hermite normal form simplices
本文研究单非平凡行 Hermite 标准形格单纯形的局部 $h^*$-多项式，证明在非对角元固定且标准化体积趋于无穷时其系数分布存在极限，并给出该极限由特定体积取值决定，同时分析了两类具体单纯形族。
In this paper, we give a formula for the area of the triangle formed by the vertices that live on a given polynomial, and we generalize this formula to the volumes of $n$-simplices with vertices on a polynomial space curve. To prove these results, we use induction arguments and a well known identity for complete symmetric polynomials.
We give an explicit upper bound on the volume of lattice simplices with fixed positive number of interior lattice points. The bound differs from the conjectural sharp upper bound only by a linear factor in the dimension. This improves significantly upon the previously best results by Pikhurko from 2001.
In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also construct families of non-congurent simplices not determined by their codimension-2 areas.
We prove sharp upper bounds on the volume and the number of lattice points on edges of higher-dimensional reflexive simplices. These convex-geometric results are derived from new number-theoretic bounds on the denominators of unit fractions summing up to one. The main algebro-geometric application is a sharp upper bound on the anticanonical degree of higher-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number one, where we completely characterize the case of equality.
Quivers constructed from hyperbolic Coxeter simplices give examples of minimal mutation-infinite quivers, however they are not the only such quivers. We classify minimal mutation-infinite quivers through a number of moves and link the representatives of the classes with the hyperbolic Coxeter simplices, plus exceptional classes which are not related to simplices.
Simplex volumes in hyperplane arrangements
该研究探讨了超平面排列中简单体积的问题，包括单位体积简单体积的最大数目、最小/最大体积简单体积的最大数目以及不同体积简单体积的排列问题。
On the geometry of stack-sorting simplices
该研究证明栈排序多面体是单纯形，给出$Ln1$排列对应多面体的相对体积为1，建立格点数量上界，并指出$2Ln1$排列生成的多面体无内点，推进了栈排序多面体的几何性质研究。
For each dimension $d$, $d$-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex …
An empty simplex is a lattice simplex with only its vertices as lattice points. Their classification in dimension three was completed by White in 1964. In dimension four, the same task was started in 1988 by Mori, Morrison, and Morrison, with their motivation coming from the close relationship between empty simplices and terminal quotient singularities. They conjectured a classification of empty simplices of prime volume, modulo finitely many exceptions. Their conjecture was proved by Sankaran (1990) with a simplified proof by Bober (2009). The same classification was claimed by Barile et al. …
We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by $n$ points in 3-space, and in general in $d$ dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by $n$ points in $\RR^3$ is at most ${2/3}n^3-O(n^2)$, and there are point sets for which this number is ${3/16}n^3-O(n^2)$. We also present an $O(n^3)$ time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every $k,d\in \NN$, $1\leq k \leq d$, the maximum number …
Simplices with fixed volumes of codimension 2 faces in a continuous deformation
针对欧氏空间中保持余维2面体积的连续形变是否必为刚体运动的开放问题，研究给出了伪欧氏空间变体情形下的反例。
A flip is a minimal move between two triangulations of a polytope. An open question is whether any two triangulations of the product of two simplices can be connected through a series of flips. This was proven in the case where one of the simplices is a triangle by Santos in 2005. In this paper we extend this to when one of the simplices is a tetrahedron.
On the volumes of simplices determined by a subset of $\mathbb{R}^d$
本文证明高维 Borel 子集确定的单纯形体积集具有正测度，并给出尖锐下界及推广的切片定理。
We investigate multi-graded Gorenstein semigroup algebras associated with an infinite family of reflexive lattice simplices. For each of these algebras, we prove that their multigraded Poincaré series is rational. Our method of proof is to produce for each algebra an explicit minimal free resolution of the ground field, in which the resolution reflects the recursive structure encoded in the denominator of the finely-graded Poincaré series. Using this resolution, we show that these algebras are not Koszul, and therefore rationality is non-trivial. Our results demonstrate how interactions betwee…
Sperner's lemma is a statement about labeled triangulations of a simplex. McLennan and Tourky (2007) provided a novel proof of Sperner's Lemma by examining volumes of simplices in a triangulation under time-linear simplex-linear deformation. We adapt a similar argument to prove Tucker's Lemma on a triangulated cross-polytope $P$. The McLennan-Tourky technique does not directly apply because this deformation may distort the volume of $P$. We remedy this by inscribing $P$ in its dual polytope, triangulating it, and considering how the volumes of deformed simplices behave.
Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in $M^d$ for $d=n$, and the volumes of $k$-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all $k$-faces; and this property still holds in $M^d$ for $d\geq n+1$ if an invariant $c_{k-1}(α^{k-1…
The cyclicity rank of empty lattice simplices
本文研究了空格点单纯形的循环秩，确定了维度 d ≤ 8 时的最大可能循环秩，并给出了该数值至对数项的渐近性态。
There exist as many index-$k$ sublattices of the hexagonal lattice up to isometry as there exist lattice triangles with normalized volume $k$ up to unimodular equivalence, which can be explained using orbifolds. In dimension 3, it was noted that the number of sublattices of the fcc and the bcc lattices and the number of lattice tetrahedra all seem to be the same. We provide a bijection between the sublattices of the coweight lattice $A^*_n$ and the $n$-dimensional lattice simplices. It explains, proves, and generalizes the observed coincidences to arbitrary dimension.
In 1982, Zaks, Perles and Wills discovered a d-dimensional lattice simplex S_{d,k} with k interior lattice points, whose volume is linear in k and doubly exponential in the dimension d. It is conjectured that, for all d \ge 3 and k \ge 1, the simplex S_{d,k} is a volume maximizer in the family P^d(k) of all d-dimensional lattice polytopes with k interior lattice points. To obtain a partial confirmation of this conjecture, one can try to verify it for a subfamily of P^d(k) that naturally contains S_{d,k} as one of the members. Currently, one does not even know whether S_{d,k} is optimal within …
Let $M^d$ be the spherical, Euclidean, or hyperbolic space of dimension $d\ge n+1$. Given any degenerate $(n+1)$-simplex $\mathbf{A}$ in $M^d$ with non-degenerate $n$-faces $F_i$, there is a natural partition of the set of $n$-faces into two subsets $X_1$ and $X_2$ such that $\sum_{X_1}V_n(F_i)=\sum_{X_2}V_n(F_i)$, except for a special spherical case where $X_2$ is the empty set and $\sum_{X_1}V_n(F_i)=V_n(\mathbb{S}^n)$ instead. For all cases, if the vertices vary smoothly in $M^d$ with a \emph{single} volume constraint that $\sum_{X_1}V_n(F_i)-\sum_{X_2}V_n(F_i)$ is preserved as a constant (…
Empty simplices of large width
本文研究空单纯形，通过引入分圆单纯形并穷举计算维数10的实例，以及利用特定循环矩阵构造任意维数的方法，首次得到了宽度大于其维数的空单纯形。
Let $\mathcal{T}^d(1)$ be the set of all $d$-dimensional simplices $T$ in $\real^d$ with integer vertices and a single integer point in the interior of $T$. It follows from a result of Hensley that $\mathcal{T}^d(1)$ is finite up to affine transformations that preserve $\mathbb{Z}^d$. It is known that, when $d$ grows, the maximum volume of the simplices $T \in \cT^d(1)$ becomes extremely large. We improve and refine bounds on the size of $T \in \mathcal{T}^d(1)$ (where by the size we mean the volume or the number of lattice points). It is shown that each $T \in \mathcal{T}^d(1)$ can be decompo…
Sylvester simplices: Triangulations and Ehrhart-theoretic aspects
本文研究 Sylvester 单纯形的三角剖分与 Ehrhart 理论性质，证明其 h* 向量单峰，确定 f* 向量部分值，并验证其在低维的 Ehrhart 魔正性。
We provide a characterization of the radii minimal projections of polytopes onto $j$-dimensional subspaces in Euclidean space $\E^n$. Applied on simplices this characterization allows to reduce the computation of an outer radius to a computation in the circumscribing case or to the computation of an outer radius of a lower-dimensional simplex. In the second part of the paper, we use this characterization to determine the sequence of outer $(n-1)$-radii of regular simplices (which are the radii of smallest enclosing cylinders). This settles a question which arose from the incidence that a paper…
The noncommutative Gurarij space $\mathbb{\mathbb{\mathbb{NG}}}$, initially defined by Oikhberg, is a canonical object in the theory of operator spaces. As the Fra\"ıssé limit of the class of finite-dimensional nuclear operator spaces, it can be seen as the noncommutative analogue of the classical Gurarij Banach space. In this paper, we prove that the automorphism group of $\mathbb{\mathbb{NG}}$ is extremely amenable, i.e.\ any of its actions on compact spaces has a fixed point. The proof relies on the Dual Ramsey Theorem, and a version of the Kechris--Pestov--Todorcevic correspondence in the …
Minimal Face Numbers for Volume Rigidity
本文推广了Maxwell的欧氏刚性边数下限，利用体积刚性拟阵与外代数移位工具，给出了纯$d$维单纯复形在$\mathbb{R}^d$中生成体积刚性泛型框架所需的最小$k$-单纯形数量，并证明了顶点移除引理。
A note on distinct volume subsets problem
本文针对不同体积子集问题及其对偶版本，构造点集改进了H_{a,d}(n)的上界，还改进了D_d(n)的上界并给出D_2(n)、D_3(n)的首个非平凡下界。
We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra.
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