线性代数网课代修|凸分析代写Convex analysis代考|MATH4071

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线性代数作业代写linear algebra代考|Helly’s Theorem

If you want to know whether a large collection of convex sets in $X=\mathbb{R}^{n}$ intersects, that is, has a common point, then you only have to check that each subcollection of $n+1$ elements has a common point.

Theorem 1.10.1 (Helly’s Theorem) A finite collection of more than $n$ convex sets in $X=\mathbb{R}^{n}$ has a common point iff each subcollection of $n+1$ sets has a common point. Example 1.10.2 (Helly’s Theorem)

1. Figure $1.21$ illustrates Helly’s theorem for four convex sets $S_{1}, S_{2}, S_{3}, S_{4}$ in the plane. These four sets have no common point, as you can check. Therefore, by Helly’s theorem, there should be a choice of three of these sets that has no common point. Indeed, one sees that $S_{1}, S_{3}, S_{4}$ have no common point.
2. Figure $1.22$ illustrates that Helly’s theorem cannot be improved: three convex subsets of the plane for which each pair has a nonempty intersection need not have a common point.

Now we prove Helly’s theorem for some special cases. It is recommended that you draw some pictures for a good understanding of the arguments that will be given.

线性代数作业代写linear algebra代考|Applications of Helly’s Theorem

Many interesting results can be proved with unexpected ease when you observe that Helly’s theorem can be applied. We formulate a number of these results. This material is optional. At a first reading, you can just look at the statements of the results below. Some of the proofs of these results are quite challenging. At the end of the chapter, hints are given.

Proposition 1.11.1 If each three points from a set in the plane $\mathbb{R}^{2}$ of at least three elements can be enclosed inside a unit disk, then the entire set can be enclosed inside a unit disk.

Here is a formulation in nontechnical language of the next result: if there is a spot of diameter $d$ on a tablecloth, then we can cover it with a circular napkin of radius $d / \sqrt{3}$

Proposition 1.11.2 (Jung’s Theorem) If the distance between any two points of a set in the plane $\mathbb{R}^{2}$ is at most 1, then the set is contained in a disk of radius $1 / \sqrt{3}$. We can also give a similar result about a disk contained in a given convex set in the plane.

Proposition 1.11.3 (Blaschke’s Theorem) Every bounded convex set $A$ in the plane of width 1 contains a disk of radius $1 / 3$ (the width is the smallest distance between parallel supporting lines; a line is called supporting if it contains a point of the set and moreover has the set entirely on one of its two sides).

线性代数作业代写linear algebra代考|Helly’s Theorem

1. 数字1.21说明了四个凸集的 Helly 定理 $S_{1}, S_{2}, S_{3}, S_{4}$ 在飞机上。这四套没有共同点， 你可以检查一下。因此，根据 Helly 定理，应该可以选择其中三个没有共同点的集 合。确实，有人看到 $S_{1}, S_{3}, S_{4}$ 没有共同点。
2. 数字 $1.22$ 说明 Helly 定理无法改进: 平面的三个凸子集 (每对具有非空交点) 不必有 公共点。 现在我们针对一些特殊情况证明 Helly 定理。建议您画一些图片，以便更好地理解将要给出 的论点。

在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions