﻿ 抽象几何代写 convex hull代写 extreme points代写 - 作业代写, 数学代写

# 抽象几何代写 convex hull代写 extreme points代写

2022-04-05 11:46 星期二 所属： 作业代写 浏览：97

## QUESTIONS 16–29

### Questions on convex sets and convex polyhedra

Question 16. Specify all the nonzero convex subsets of the line R1.

Question 17. 1) Assume that any two vertices of a convex polygon ∆ R2 are joint by a side of ∆. Show that the polygon is a triangle.

2) Assume that any two vertices of a convex polyhedron ∆ R2are joint by an edge of ∆. Show that the polyhedron is a tetrahedron.

Question 18. Let A be a point belonging to a face F of a convex polyhedron ∆. Assume that A is the center of positive masses located at some vertices A1, . . . , Ak of ∆. Prove that A1, . . . , Ak belong to the face F.

### Questions on convex hull  抽象几何代写

Question 19. Let Γ R2 be the circle x2 + y2 = 1;

Let Γ1 R2 be a half-circle x2 + y2 = 1, y 0

and let A, B be the points A = (1, 0), B = (1, 0).

Find the convex hulls of the following sets:

1) of the circle Γ;

2) of the half-circle Γ1;

3) of the half-circle without one endpoint Γ1 \ {A};

4) of the half-circle without two endpoint Γ1 \ ({A} ∪ {B};

Radon’s theorem for Rd in exercises. In the d-dimensional space Rd Radon’s theorem tells that if S is a set containing at least n points in Rd and n d + 2, then we can subdivide S to two subsets so that their convex hulls intersect. Prove the theorem following the outlined exercises:

Question 21. If there is a collection of d + 1 points from S that lie on a (d 1)– dimensional hyperplane, reduce the problem to (d  1)-dimensional Radon theorem.

Question 22. Show it is enough to prove the theorem for sets containing n = d+ 2 points.

Question 23. Suppose that S consists of d+ 2 points p1, …, pd+2 and every collection of d+ 1 of them doesn’t lie on one hyperplane. Show that we can find non-zero numbers λ1, …, λn+1 with sum 1, so that

pd+2 = λ1p1 + + λd+1pd+1.

Question 25. Deduce Radon’s theorem from the result of Question 24.

### Helly’s theorem for Rdin exercise.

Question 26. Deduce Helly’s theorem in d-dimensional space Rd from Radon’s theorem proved in the exercise above. Namely show that if every d + 1 sets in a finite collection of convex sets intersect, then all the sets in this collection intersect.

Hint: the arguments are very close to those used in the deduction of planar Helly’s theorem from planar Radon’s theorem.

### A couple of questions below are related to Helly’s theorem  抽象几何代写

Question 27 Let T1, . . . , Tm R2 be m 3 vertical segments (i.e. segments

which are parallel to the y-coordinate line). Assume that foor any three segments there is a line that intersects all of them. Prove that there exits a line that intersects all segments.

Hint: Any non vertical line ax + b can be identified with the point (a, b) on a plane. With any segment T i associate a set G i of lines intersecting T i. Try to apply Helly’s theorem for the set {Gi} on plane.

Question 28. Prove that if every three points of a finite collection of points in a plane lie inside some circle of radius one, then all the points of this collection lie inside some circle of radius one.

Hint: Consider the set of circles of radius one with centers at the points from our collection of points. Try to apply Helly’s theorem for the set of circles on plane.

### A question on extreme points  抽象几何代写

Question 29. An extreme point in a convex set ∆ is a point p which does not lie in any open line segment joining two points in ∆. Prove that p is an extreme point if and only if the set ∆ \ {p} is convex.