By Chuanming Zong (auth.), James J. Dudziak (eds.)

Convex and discrete geometry is without doubt one of the such a lot intuitive matters in arithmetic. you can still clarify a lot of its difficulties, even the main tough - reminiscent of the sphere-packing challenge (what is the densest attainable association of spheres in an n-dimensional space?) and the Borsuk challenge (is it attainable to partition any bounded set in an n-dimensional house into n+1 subsets, each one of that's strictly smaller in "extent" than the total set?) - in phrases layman can comprehend; and you can kind of make conjectures approximately their recommendations with little education in mathematics.

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The Venkov-McMullen Theorem 43 §3. A. Venkov [1], P. McMullen [2]). Every convex tmnslative tile is also a lattice tile. As preparation for the proof of this theorem, let us introduce some notation and definitions. For P an n-dimensional convex polytope, let Ff, F~, ... , F~(P,i) denote the i-dimensional faces of P. When all the facets F;,-l, F;-l, . , F;(P~n_l) as well as the polytope P itself are centrally symmetric, We call the collection Fi of those facets which contain a translate of a 2 a belt of P.

4 Local Packing Phenomena §l. Introduction Let K be a fixed convex body in Rn. We call the largest number of nonoverlapping translates of K which can be brought into contact with K the kissing number of K and denote it by h(K). A closely related but contrasting concept is the blocking number of K, denoted z(K), which is the smallest number of nonoverlapping translates of K which are in contact with K and prevent any other translate of K from touching K. 1. Let Kl and K2 be two distinct convex bodies in Rn.

Since g( u) was any continuous function, it follows that lim G1(U) 1-++00 = G(U). 8) For an arbitrary u E 8(B), define Tu(v) = max{O, (u,v)}. Obviously, Tu(v) is positive on the open hemisphere B: = {x E 8(B) : (x, u) > O} and vanishes on the rest of 8(B). 2) in mind, it can be shown that there exists a positive c > 0 such that for all u E 8(B), r la(B) Tu(v)G(dv) ~ 2c. 42 3. The Venkov-McMullen Theorem and Stein's Phenomenon Also, since {Tu(v) : u E 8(B)} is trivially equicontinuous, lim r 1->+00 18(B) Tu(V)Gl(dv) = r 18(B) Tu(v)G(dv) uniformly in u.