Related papers: Fat Shellable Spheres
We introduce the fatness parameter of a 4-dimensional polytope P, defined as \phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes bounded? We…
In this paper, we discuss f- and flag-vectors of 4-dimensional convex polytopes and cellular 3-spheres. We put forward two crucial parameters of fatness and complexity: Fatness F(P) := (f_1+f_2-20)/(f_0+f_3-10) is large if there are many…
A family of potential-density pairs that represent spherical shells with finite thickness is obtained from the superposition of spheres with finite radii. Other families of shells with infinite thickness with a central hole are obtained by…
Nevo, Santos, and Wilson constructed $2^{\Omega(N^d)}$ combinatorially distinct simplicial $(2d-1)$-spheres with $N$ vertices. We prove that all spheres produced by one of their methods are shellable. Combining this with prior results of…
Given R\subset N, an (R,k)$-sphere is a k-regular map on the sphere whose faces have gonalities i\in R. The most interesting/useful are (geometric) fullerenes, i.e., (\{5,6\},3)$-spheres. Call \kappa_i=1 + \frac{i}{k} - \frac{i}{2} the…
Soliton spheres are immersed 2-spheres in the conformal 4-sphere S^4=HP^1 that allow rational, conformal parametrizations f:CP^1->HP^1 obtained via twistor projection and dualization from rational curves in CP^{2n+1}. Soliton spheres can be…
We construct 2^{\Omega(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves…
We construct a 2-parameter family of 4-dimensional polytopes with extreme combinatorial structure: In this family, the ``fatness'' of the f-vector gets arbitrarily close to 9, the ``complexity'' (given by the flag vector) gets arbitrarily…
In connection with commutative algebra, Bayer et al. introduced cut complexes in [Topology of cut complexes of graphs, SIAM J.\ Discrete Math., 38(2):1630-1675, 2024]. For a positive integer $k$, the $k$-cut complex of a graph $G$, denoted…
In 1964 A. Bruckner observed that any bounded open set in the plane has an inscribed triangle, that is a triangle contained in the open set and with the vertices lying on the boundary. We prove that this triangle can be taken uniformly fat,…
We define $(\alpha_n)$ -regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the…
It is known that the $(2k-1)$-sphere has at most $2^{O(n^k \log n)}$ combinatorially distinct triangulations with $n$ vertices, for every $k\ge 2$. Here we construct at least $2^{\Omega(n^k)}$ such triangulations, improving on the previous…
There is a two-component log-gas system with Boltzmann factor which provides an interpolation between the eigenvalue PDF for $\beta = 1$ and $\beta = 4$ invariant random matrix ensembles. The solvability of this log-gas system relies on the…
We construct a new 2-parameter family E_mn of self-dual 2-simple and 2-simplicial 4-polytopes, with flexible geometric realisations. E_44 is the 24-cell. For large m,n the f-vectors have ``fatness'' close to 6. The E_t-construction of…
We explore some generalizations of fullerenes F_v (simple polyhedra with v vertices and only 5- and 6-gonal faces) seen as (d-1)-dimensional simple manifolds (preferably, spherical or polytopal) with only 5- and 6-gonal 2-faces. First,…
The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take…
We present a new algorithmic approach that can be used to determine whether a given quadruple $(f_0,f_1,f_2,f_3)$ is the f-vector of any convex 4-dimensional polytope. By implementing this approach, we classify the f-vectors of 4-polytopes…
The purpose of this paper is to construct some special kind of subschemes in $\mathbb{P}^N$ with $ N\ge 3$, which we call them "fat flat subschemes" and compute their Waldschmidt constants. These subschemes are constructed by adding, in a…
We calculate the large-$N$ expansion of the sphere free energy $F=-\log Z_{S^d}$ of the O(N) $\phi^4$ and the Gross-Neveu $(\bar{\psi} \psi)^2$ CFTs to order $1/N$. Analytic regularization of these theories requires consistently shifting…
A derivation of the Ces\`aro-Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-linear boundaries.…