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We find an optimal upper bound on the volume of the John ellipsoid of a k-dimensional section of the n-dimensional cube, and an optimal lower bound on the volume of the Lowner ellipsoid of a projection of the n-dimensional cross-polytope onto a k-dimensional subspace, which are respectively $$\bigl (\frac{n}{k}\bigr )^{{k}/{2}}$$ and $$\bigl (\frac{k}{n}\bigr )^{{k}/{2}}$$ of the volume of the unit ball in $$\mathbb {R}^k$$. Also, we describe all possible vectors in $$\mathbb {R}^n,$$ whose coordinates are the squared lengths of a projection of the standard basis in $$\mathbb {R}^n$$ onto a k-dimensional subspace.

Consider an $$n \times n \times n$$ cube Q consisting of $$n^3$$ unit cubes. A tripod of order n is obtained by taking the $$3n-2$$ unit cubes along three mutually adjacent edges of Q. The unit cube corresponding to the vertex of Q where the edges meet is called the center cube of the tripod. The function f(n) is defined as the largest number of integral translates of such a tripod that have disjoint interiors and whose center cubes coincide with unit cubes of Q. The value of f(n) has earlier been determined for $$n \le 9.$$ The function f(n) is here studied in the framework of the maximum clique problem, and the values $$f(10) = 32$$ and $$f(11)=38$$ are obtained computationally. Moreover, by prescribing symmetries, constructive lower bounds on f(n) are obtained for $$n \le 26.$$ A conjecture that f(n) is always attained by a packing with a symmetry of order 3 that rotates Q around the axis through two opposite vertices is disproved.

This paper studies random cubical sets in $\mathbb{R}^d$. Given a cubical set $X\subset \mathbb{R}^d$, a random variable $\omega_Q\in[0,1]$ is assigned for each elementary cube $Q$ in $X$, and a random cubical set $X(t)$ is defined by the sublevel set of $X$ consisting of elementary cubes with $\omega_Q\leq t$ for each $t\in[0,1]$. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown., Comment: 19 pages, 14 figures

We establish an isoperimetric inequality with constraint by $$n$$n-dimensional lattices. We prove that, among all sets which consist of lattice translations of a given rectangular parallelepiped, a cube is the best shape to minimize the ratio involving its perimeter and volume as long as the cube is realizable by the lattice. For its proof a solvability of finite difference Poisson---Neumann problems is verified. Our approach to the isoperimetric inequality is based on the technique used in a proof of the Aleksandrov---Bakelman---Pucci maximum principle, which was originally proposed by Cabre (Butll Soc Catalana Mat 15:7---27, 2000) to prove the classical isoperimetric inequality.

Cube tilings formed by $$n$$ -dimensional $$4\mathbb Z ^n$$ -periodic hypercubes with side $$2$$ and integer coordinates are considered here. By representing the problem of finding such cube tilings within the framework of exact cover and using canonical augmentation, pairwise nonisomorphic 5-dimensional cube tilings are exhaustively enumerated in a constructive manner. There are 899,710,227 isomorphism classes of such tilings, and the total number of tilings is 638,560,878,292,512. It is further shown that starting from a 5-dimensional cube tiling and using a sequence of switching operations, it is possible to generate any other cube tiling.

An n-dimensional cross consists of 2n+1 unit cubes: the “central” cube and reflections in all its faces. A tiling by crosses is called a Z-tiling if each cross is centered at a point with integer coordinates. Periodic tilings of ℝ n by crosses have been constructed by several authors for all n∈N. No non-periodic tiling of ℝ n by crosses has been found so far. We prove that if 2n+1 is not a prime, then the total number of non-periodic Z-tilings of ℝ n by crosses is $2^{\aleph _{0}}$ while the total number of periodic Z-tilings is only ℵ0. In a sharp contrast to this result we show that any two tilings of ℝ n ,n=2,3, by crosses are congruent. We conjecture that this is the case not only for n=2,3, but for all n where 2n+1 is a prime.

We show that the Banach---Mazur distance from any centrally symmetric convex body in źn to the n-dimensional cube is at most $$\sqrt{n^2-2n+2+\frac{2}{\sqrt{n+2}-1}},$$ which improves previously known estimates for "small" nź3. (For large n, asymptotically better bounds are known; in the asymmetric case, exact bounds are known.) The proof of our estimate uses an idea of Lassak and the existence of two nearly orthogonal contact points in John's decomposition of the identity. Our estimate on such contact points is closely connected to a well-known estimate of Gerzon on equiangular systems of lines.

We exhibit a nonregular triangulation for the product of two tetrahedra. This answers a question by Gel'fand, Kapranov, and Zelevinsky. We also give a complete classification of the symmetry classes of regular triangulations of ź2ן3. Our nonregular triangulation of ź3ן3 can be extended to a nonregular triangulation of the six-dimensional cube. The four-dimensional cube is the smallest cube with a nonregular triangulation.

We consider two on-line methods of covering the unit cube of Euclideand-space by sequences of cubes. The on-line restriction means that we are given the next cube from the sequence only after the preceding cube has been put in place without the possibility of changing the placement. The first method enables on-line covering of the unit cube by an arbitrary sequence of cubes whose total volume is at least 3...2d?4. The second method is more complicated, but, asymptotically, asd tends to infinity, it yields an efficiency of the order of magnitude 2d with factor 1. So, asymptotically, it is as good as the best possible non-on-line method of covering the unit cube by cubes.

Families of nonlattice tilings of ?n by unit cubes are constructed. These tilings are specializations of certain families of nonlinear codes overGF(2). These cube-tilings provide building blocks for the construction of cube-tilings such that no two cubes have a high-dimensional face in common. We construct cube-tilings of ?n such that no two cubes have a common face of dimension exceeding % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgk% HiTmaaleaaleaacaaIXaaabaGaaG4maaaakmaakaaabaGaamOBaaWc% beaaaaa!3A59! $$n - \tfrac{1}{3}\sqrt n$$ .

We discuss packings of sequences of convex bodies of Euclideann-spaceEn in a box and particularly in a cube. Following an Auerbach-Banach-Mazur-Ulam problem from the well-knownScottish Book, results of this kind are called potato-sack theorems. We consider on-line packing methods which work under the restriction that during the packing process we are given each succeeding "potato" only when the preceding one has been packed. One of our on-line methods enables us to pack into the cube of sided>1 inEn every sequence of convex bodies of diameters at most 1 whose total volume does not exceed ( $$(d - 1)(\sqrt d - 1)^{2(n - 1)} /n!.$$ ). Asymptotically, asd??, this volume is as good as that given by the non-on-line methods previously known.

We consider how to map the sites of a square region of planar lattice into a three-dimensional cube, so as to minimize the maximum distortion of distance. We consider the cube to be endowed with a "foliated" geometry in which horizontal distance is standard but vertical communication only occurs at the surface of the cube. These geometries may naturally arise if a planar data set is to be stored in a stack of chips. It is proved that any one-to-one map which fills the cube with a fixed "density" must produce a distortion of distance which grows as the one-sixth power of the diameter of the square and the two-thirds power of the density. Moreover, we explicitly define one-to-one maps with 100% density, one-sixth power stretching, and a small leading coefficient. As a final note, a high-dimensional analog is considered.