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Register a new userNew upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry
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In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zongs recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840ldots$ to $0.3745ldots$, getting closer to the best known lower bound of $0.3673ldots$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.
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This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in three-dimensional space, it must be a parallelohedron. In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron.
This paper proves the following statement: {it If a convex body can form a twofold translative tiling in $mathbb{E}^3$, it must be a parallelohedron.} In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron.
We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${mathbb R}^n$ with $0in {rm int}(K)$ and if $mu $ is a measure on ${mathbb R}^n$ with a locally integrable non-negative density $g$ on ${mathbb R}^n$, then begin{equation*}mu (K)leq left (csqrt{n-k}right )^kmax_{Fin G_{n,n-k}}mu (Kcap F)cdot |K|^{frac{k}{n}}end{equation*} for every $1leq kleq n-1$. Also, if $mu $ is even and log-concave, and if $K$ is a symmetric convex body in ${mathbb R}^n$ and $D$ is a compact subset of ${mathbb R}^n$ such that $mu (Kcap F)leq mu (Dcap F)$ for all $Fin G_{n,n-k}$, then begin{equation*}mu (K)leq left (ckL_{n-k}right )^{k}mu (D),end{equation*} where $L_s$ is the maximal isotropic constant of a convex body in ${mathbb R}^s$. Our method employs a generalized Blaschke-Petkantschin formula and estimates for the dual affine quermassintegrals.
In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p in (1, 1.58]$. We conjecture that the family also exists for $p in (1.58, log_2 3 = 1.5849625ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.
We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the other conditions). We show that any set inner product can be embedded into an inner product space on the associated support functions, thereby extending fundamental results of Hormander and Radstrom. The set inner product provides a geometry on the space of convex bodies. We explore some of the properties of that geometry, and discuss an application of these ideas to the reconstruction of ancestral ecological niches in evolutionary biology.
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