Prove that there exist two lines

Reference no: EM131085722

Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-

(a) Suppose X₁, X₂, . . . , X_n ⊂ R² are convex sets such that through every four of them there exists a line intersecting them. Prove that there exist two lines l₁ and l₂ such that every X_iintersects l₁ or l₂.

(b) Let X₁, . . . , X_n ⊂ R^d be full dimensional rectangular prisms with faces parallel to the coordinate axes. Determine, which proof, a function f(d), dependent only on d, such that the following statement is true:

"Suppose that for any f(d) of these convex sets, there exists a hyperplane intersecting them. Then there exists a hyperplane intersecting all X_i, 1 ≤ i ≤ n.", or prove such a function cannot exist.

Reference no: EM131085722

Questions Cloud

What is the most applicable route in the present : What is the most applicable route in the present?

Recommendations add value to the areas of people : Continuing to use the SHRM case study presented in the classroom, students will present a second power point presentation that builds on the information from the first presentation and applies some of the additional material assigned in the class.

What lessons can be derived from boserup''s analysis : What lessons can be derived from Boserup's analysis?

Himss davies award case study : The student will choose a 2014 HIMSS Davies Award recipient, then they will read the related case study. After careful consideration, the student will answer the following questions:

Prove that there exist two lines : Suppose X1, X2, . . . , Xn ⊂ R2 are convex sets such that through every four of them there exists a line intersecting them. Prove that there exist two lines l1 and l2 such that every Xi intersects l1 or l2

What are the defining characteristics of green revolution : What are the defining characteristics of the ‘Green Revolution'?

Do you think voter id laws are necessary : Do you think voter ID laws are necessary? Why or why not. Do you believe that voter ID laws disenfranchise voters who are disproportionately members of minority groups or the poor? Why or why not?

Completeness of the surgical informed consent : Data have been collected on the completeness of the surgical informed consent. More than 200 records have been reviewed for the presence of the following data items on the consent form:

Prove that m is a matroid : Let M = (S, I) be a matroid. Prove that M∗ is a matroid. Prove that the rank function r∗ of M∗ is given by r∗(A) = |A| - r(M) + r(S\A), and conclude TM∗ (x, y) = TM(y, x)

User Account

All Pages