Question

Let \(\left\\{K_{i}\right\\}\) be a finite family of convex compact sets in \(\mathbf{R}^{n}\) such that every subfamily consisting of \(n+1\) members has a nonempty intersection. Helly's theorem asserts that then \(\bigcap K_{i} \neq \emptyset\). Prove it for \(n=1\). Show an example for \(n=2\) that \(n+1\) is necessary. Hint: Consider the interval between the maximum of the left endpoints of \(K_{i}\) and the minimum of the right endpoints of \(K_{i}\). For the general case, see for example [DGK]. Example: three lines forming a triangle. 1.60 Let \(X\) be a Banach space. Show that if \(A \subset X\) is totally bounded, then there is a sequence \(\left\\{x_{n}\right\\} \in X\) such that \(x_{n} \rightarrow 0\) in \(X\) and \(A \subset\) sconv \(\left\\{x_{n}\right\\}\) (see Exercise 1.7). In particular, for every compact subset \(A\) of \(X\) there exists a sequence \(\left\\{x_{n}\right\\}\) such that \(x_{n} \rightarrow 0\) and \(A \subset \overline{\operatorname{conv}}\left\\{x_{n}\right\\}\) (Grothendieck). Hint: We set \(A_{1}=A\), let \(B_{1}\) be a finite \(2^{-2}\) -net in \(A_{1}\). If \(A_{i}\) and \(B_{i}\) were defined for \(i \leq n\), let \(A_{n+1}=\left(A_{n}-B_{n}\right) \cap 2^{-2 n} B_{X} ;\) note that every \(a_{n} \in A_{n}\) is of the form \(a_{n}=a_{n+1}+b_{n}\), where \(a_{n+1} \in A_{n+1}, b_{n} \in B_{n} .\) Let \(B_{n+1}\) be a finite \(2^{-2 n}\) -net in \(A_{n+1}\). Therefore, every element \(a \in A=A_{1}\) is of the form \(a=b_{1}+a_{2}=b_{1}+b_{2}+a_{3}=\ldots=\sum_{1}^{n} b_{n}+a_{n+1} .\) Since \(a_{n+1} \in 2^{-2 n} B_{X}\)

Short answer

\(\bigcap K_i eq \emptyset \) when every subfamily of n+1 sets intersects for n=1. An example for n=2 shows \(n+1\) members is necessary.

Step-by-step solution

Prove Helly's theorem for n=1

Consider each set in \{K_i\} to be an interval on the real line, \[K_i = [a_i, b_i]\]. Every subfamily consisting of 2 members (n+1 for n=1) has a nonempty intersection, meaning \[\forall i, j, [a_i, b_i] \cap [a_j, b_j] eq \emptyset.\] Therefore, the intersection must include an interval between the maximum of all left endpoints and the minimum of all right endpoints, \[\bigcap K_{i} = [\max(a_i), \min(b_i)] eq \emptyset.\]

Show an example for n=2 where n+1 is necessary

Consider three lines forming a triangle in \mathbf{R}^2\, such as: \[K_1 = L_1, K_2 = L_2, K_3 = L_3\] where \L_1, L_2,\ and \L_3\ are line segments that intersect pairwise but not all three together. In this example, every pair of \K_i\ and \K_j\ has an intersection, but the intersection of all three is empty. This demonstrates that for n=2, n+1 is necessary.

Properties of totally bounded sets in Banach spaces

Given a totally bounded set \(A \subset X\), construct a sequence that approximates elements in A. Start with \A_1 = A\ and pick a finite \2^{-2}\ Net \B_1\ in \A_1\. For subsequent sets \A_n\, choose a finite \2^{-2n}\-Net \B_n\. Then, \A_{n+1} = (A_n - B_n) \cap 2^{-2n} B_X\, ensuring every \a_n \in A_n\ is of the form \a_n = a_{n+1} + b_n\. Continue iterating to achieve \A \subset \text{sconv}\{x_n\},'s closure, showing that there exists a sequence \{x_n\} where \ x_n \rightarrow 0\.

Key concepts

Helly's Theorem

Helly's Theorem is an important result in convex geometry. It states that for a finite family of convex compact sets in \(\textbf{R}^n\), if every subfamily of at most \(n+1\) sets has a non-empty intersection, then the whole family has a non-empty intersection. This theorem is particularly useful in higher-dimensional geometry and has various applications.
To understand it better, let's consider a simple case when \(n=1\). Here, the sets are intervals on the real line, denoted as \([a_i, b_i]\). According to Helly's Theorem, if every pair of these intervals intersects, there must be a common point that belongs to all intervals.

Banach Spaces

A Banach space is a complete normed vector space. This means that it is a vector space equipped with a norm, and it is complete in the sense that every Cauchy sequence in the space converges to a point within the space.
An example is the space of continuous functions on a closed interval \([a, b]\)\, denoted \(C([a, b])\), with the supremum norm. Banach spaces are fundamental in functional analysis and are used to study various problems in mathematical analysis and its applications.

Totally Bounded Sets

A set \(A\) in a metric space is totally bounded if, for any \(\text{ε}>0\), we can cover \(A\) with a finite number of subsets whose diameter is less than \(ε\).
In simpler terms, a set is totally bounded if it can be covered by a finite number of balls of any chosen small radius. This concept is crucial in analysis as it relates to compactness—a compact set is always totally bounded and closed.

Convex Sets

A set \(K\) in a vector space is convex if, for any two points \(x, y \in K\), the line segment joining \(x\) and \(y\) lies entirely within \(K\). Mathematically, this means that for any \(\text{λ} \in [0,1]\),\[ \text{λ}x+(1- \text{λ})y \in K.\]\
Convex sets are fundamental in various fields such as optimization, economics, and geometry. For example, the feasible region defined by linear inequalities in optimization problems is a convex set.

Infinite Dimensional Geometry

Infinite dimensional geometry involves studying the geometric properties of spaces that have infinitely many dimensions. These spaces are usually Banach or Hilbert spaces and are used in many areas like quantum mechanics, PDEs, and more.
One crucial aspect is understanding how geometric intuition from finite dimensions extends or fails in infinite dimensions. For example, the unit ball in an infinite-dimensional Banach space can be arbitrarily 'spiky', unlike in finite dimensions where it is always 'smooth'.