抽象几何代写 convex hull代写 extreme points代写

QUESTIONS 16–29

抽象几何代写 Questions on convex sets and convex polyhedra Question 16. Specify all the nonzero convex subsets of the line R1. Question 17. 1)

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, . . . , A_k of ∆. Prove that A₁, . . . , A_k 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 R^d in exercises. In the d-dimensional space R^d Radon’s theorem tells that if S is a set containing at least n points in R^d 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, …, p_d₊₂ and every collection of d+ 1 of them doesn’t lie on one hyperplane. Show that we can find non-zero numbers λ₁, …, λ_n₊₁ with sum 1, so that

p_d₊₂ = λ₁p₁ + … + λ_d₊₁p_d₊₁.

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

Helly’s theorem for R^d in exercise.

Question 26. Deduce Helly’s theorem in d-dimensional space R^d 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, . . . , T_m ⊂ 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 {G_i} 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.

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