Convex bodies: the brunn–minkowski theory

Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

Introduction 1. Packaging and covering densities 2. The existence of reasonably dense packings 3. The existence of reasonably economical coverings 4. The existence of reasonably dense lattice packings 5. The existence of reasonably economical lattice coverings 6. Packings of simplices cannot be very dense 8. Coverings with spheres cannot be very economical Bibliography Index.

Packing and Covering. By C. A. Rogers. Pp. viii, 111. 30s. 1964. (Cambridge)

Reconfigurable, ordered matter offers great potential for future low-power computer memory by storing information in energetically stable configurations. Among these, skyrmions—which are topologically protected, robust excitations that have been demonstrated in chiral magnets1–4 and in liquid crystals5–7—are driving much excitement about potential spintronic applications8. These information-encoding structures topologically resemble field configurations in many other branches of physics and have a rich history9, although chiral condensed-matter systems so far have yielded realizations only of elementary full and fractional skyrmions. Here we describe stable, high-degree multi-skyrmion configurations where an arbitrary number of antiskyrmions are contained within a larger skyrmion. We call these structures skyrmion bags. We demonstrate them experimentally and numerically in liquid crystals and numerically in micromagnetic simulations either without or with magnetostatic effects. We find that skyrmion bags act like single skyrmions in pairwise interaction and under the influence of current in magnetic materials, and are thus an exciting proposition for topological magnetic storage and logic devices. Structures containing multiple skyrmions inside a larger skyrmion—called skyrmion bags—are experimentally created in liquid crystals and theoretically predicted in magnetic materials. These may have applications in information storage technology.

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Two-dimensional skyrmion bags in liquid crystals and ferromagnets

It is proved that if $C$ is a convex body in ${\Bbb R}^n$ then $C$ has an affine image $\widetilde C$ (of non-zero volume) so that if $P$ is any 1-codimensional orthogonal projection, 
$$|P\widetilde C| \ge |\widetilde C|^{n-1\over n}.$$ 
It is also shown that there is a pathological body, $K$, all of whose orthogonal projections have volume about $\sqrt{n}$ times as large as $|K|^{n-1\over n}$.

Shadows of convex bodies

The densest packings of n congruent circles in a circle are known for n 12 and n = 19. In this article we examine the case of 13 congruent circles. We show that the optimal configurations are identical to Kravitz's conjecture. We use a technique developed from a method of Bateman and Erd˝os, which proved fruitful in investigating the cases n = 12 and 19. MSC 2000: 52C15 OP, where O is the origin. For a point P and a vector ~a by P +~a we always mean the vector ~ OP +~a. In this paragraph we give a brief account of the history of the subject. This review is based on the doctoral dissertation of Melissen (13). The problem of finding the densest packing of congruent circles in a circle arose in the 1960s. The question was to find the smallest circle in which we can pack n congruent unit circles, or equivalently, the smallest circle in which we can place n points with mutual distances at least 1. Dense circle packings were first given

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The densest packing of 13 congruent circles in a circle.

Dense packings of n congruent circles in a circle were given by Kravitz in 1967 for n = 2,..., 16. In 1969 Pirl found the optimal packings for n ≤ 10, he also conjectured the dense configurations for 11 ≤ n ≤ 19. In 1994, Melissen provided a proof for n = 11. In this paper we exhibit the densest packing of 19 congruent circles in a circle with the help of a technique developed by Bateman and Erdos.

The Densest Packing of 19 Congruent Circles in a Circle

The Lp dual Minkowski problem for p > 1 and q > 0

Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by K-n the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sib intrinsic volumes V-s(K-n) of K-n for s is an element of {1, ... , d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of K-n. The essential tools are the economic cap covering theorem of Barany and Larman, and the Efron-Stein jackknife inequality.

/pdf/intrinsic-volumes-of-inscribed-random-polytopes-in-smooth-4eskrfuz2b.pdf

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of Barany and Larman, and the Efron-Stein jackknife inequality.

/pdf/intrinsic-volumes-of-inscribed-random-polytopes-in-smooth-2vjrpib2i4.pdf

Ferenc Fodor

Papers

The densest packing of 13 congruent circles in a circle.

The Densest Packing of 19 Congruent Circles in a Circle

The Lp dual Minkowski problem for p > 1 and q > 0

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Intrinsic volumes of inscribed random polytopes in smooth convex bodies