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Convex and discrete geometry is one of the most intuitive subjects in mathematics. One can explain many of its problems, even the most difficult - such as the sphere-packing problem (what is the densest possible arrangement of spheres in an n-dimensional space?) and the Borsuk problem (is it possible to partition any bounded set in an n-dimensional space into n+1 subsets, each of which is strictly smaller in "extent" than the full set?) - in terms that a layman can understand; and one can reasonably make conjectures about their solutions with little training in mathematics.



1. Borsuk’s Problem

Let X denote a subset of R n . As usual, we call
$$d\left( X \right) = \mathop {\sup }\limits_{x,y \in X} \left\| {x - y} \right\|$$
the diameter of X. In studying the relation between a set and its subsets of smaller diameter, K. Borsuk [2] in 1933 raised the following famous problem:
Borsuk’s Problem. Is it true that every bounded set X in R n can be partitioned into n +1 subsets X1, X2,...., Xn+1 such that
$$d\left( {{X_i}} \right) < d\left( X \right), i = 1,2, \ldots ,n + 1?$$
Chuanming Zong, James J. Dudziak

2. Finite Packing Problems

In n-dimensional Euclidean space, how should one arrange m nonoverlapping translates of a given convex body K in order to minimize the diameter, the surface area, or the volume of their convex hull?
Chuanming Zong, James J. Dudziak

3. The Venkov-McMullen Theorem and Stein’s Phenomenon

Let M be an n-dimensional compact set with interior points. If there exists a set of points X such that
$$\bigcup\limits_{x \in X} {\left( {M + x} \right)} = {R^n}$$
$$\left( {\operatorname{int} \left( M \right) + {x_1}} \right) \cap \left( {\operatorname{int} \left( M \right) + {x_2}} \right) = 0$$
whenever x1, x2X with x1x2, then we call M a translative tile. In addition, if X is a lattice in R n , then we call M a lattice tile.
Chuanming Zong, James J. Dudziak

4. Local Packing Phenomena

Let K be a fixed convex body in R n . 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. Concerning kissing numbers and blocking numbers, one can raise the following intuitive problem:
Problem 4.1. Let K1 and K2 be two distinct convex bodies in R n . Does h(K1 < h(K2) always imply that z(K1) ≤ z(K2)?
Chuanming Zong, James J. Dudziak

5. Category Phenomena

In a metric space {ℜ, ρ}, a subset is called meager or of the first category if it can be represented as a countable union of nowhere dense subsets. We say that a property holds for most1 elements of Ii if it holds for all elements of ℜ that lie off a meager subset. In 1899, R. Baire [1] found that every meager subset of a compact metric space or a locally compact metric space has a dense complement. So, in a topological sense, meager sets are “small,” whereas their complements are “large.”
Chuanming Zong, James J. Dudziak

6. The Busemann-Petty Problem

The search for relationships between a convex body and its projections or sections has a long history. In 1841, A. Cauchy found that the surface area of a convex body can be expressed in terms of the areas of its projections as follows:
$$s\left( K \right) = \frac{1}{{{\omega _{n - 1}}}}\int_{\partial \left( B \right)} {\bar v\left( {{P_u}\left( K \right)} \right)d\lambda \left( u \right)} .$$
Here, s(K) denotes the surface area of a convex body KR n , \(\bar v\left( X \right)\) denotes the (n − 1)-dimensional “area” of a set XR n −1, P u denotes the orthogonal projection from R n to the hyperplane H u = {xR n : 〈x, u〉 = 0} determined by a unit vector u of R n , and λ denotes surface-area measure on (B). In contrast, the closely related problem of finding an expression for the volume of K in terms of the areas of its projections P u (K) (or the areas of its sections I u (K) = KH u ) proved to be unexpectedly and extremely difficult.
Chuanming Zong, James J. Dudziak

7. Dvoretzky’s Theorem

Definition 7.1. Given a number є strictly between 0 and 1, an n-dimensional convex set K is called an є-sphere if there exists a positive number r such that
$$rB \subseteq K \subseteq r\left( {1 + \varepsilon } \right)B.$$
Chuanming Zong, James J. Dudziak


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