Determining the metric space recall

Reference no: EM13990823

1. For any set F and point x in a metric space recall that d(x, F ) := inf{d(x, y): y ∈ F }. Let F be a closed set in a normed linear space S with the usual distance d(x, y) := x - y . Show that d(·, F ) isa convex function if and only if F is a convex set.

2. Let U be a convex open set in Rk . For any set A ⊂ Rk let - A := {-x : x ∈ A}. Assume U = -U , so that 0 ∈ U . Let µ be a measure defined on the Borel subsets of U with 0 <>(U ) <>∞ and µ(B) ≡ µ(-B). Let f be a convex function on U . Show that f (0) ≤ ( f dµ/µ(U ). Hint: Use the image measure theorem 4.2.8 with T (x ) ≡ -x.

Reference no: EM13990823

Questions Cloud

Calculate the total variable and total fixed cost : Calculate the total variable and total fixed cost of the hotel for this year - Identify and state the major factors that a bank would take into account before deciding whether to grant an increase in the overdraft of a business.

What is the potential energy of the mass : A mass of 3.0 kg is subject to a force F(x) = 8.0 N - (4.0 N/m)x The potential energy of the mass is zero at x = 0. What is the potential energy of the mass at x = 2.0 m?

Nondecreasing function defined on an interval : Let f be a convex function defined on a convex set A ⊂ Rk . For some fixed x and y in Rk let g(t ) := f (x + ty) + f (x - ty) whenever this is defined. Show that g is a nondecreasing function defined on an interval (possibly empty) in R.

What is the fastest speed that the car can safely make turn : A 1000kg car goes around a sharp turn of radius of curvature of 25.0m. Given that the coefficient of friction between the tires and the roadway is 0.550. What is the fastest speed that the car can safely make the turn?

Determining the metric space recall : For any set F and point x in a metric space recall that d(x, F ) := inf{d(x, y): y ∈ F }. Let F be a closed set in a normed linear space S with the usual distance d(x, y) := x - y . Show that d(·, F ) isa convex function ..

Determine the spring constant for each spring : The graph below shows the stretching of two different springs A and B, when different forces were applied. Which spring is stiffer? That is, which spring requires a larger pull to get the same amount of stretch? Explain how you can tell by looking ..

Finite dimensional banach space : 1. Show that in any finite dimensional Banach space (Rk with any norm), for any closed, convex set C and any point x not in C , there is at least one nearest point y in C ; in other words, ly - x l= infz∈C lz - x l.

Which of the following are measures of variability : Which of the following are measures of central tendency? Select all that apply

Because of such relationships : Let (X, l·l) be a real normed space, E a linear subspace, and h ∈ E t. Give a proof that h can be extended to be a member of X t (the Hahn-Banach the- orem, 6.1.4) based on Theorem 6.2.11. Hint: Let U := {x ∈ X : lx l 1}. (Because of such relation..

User Account

All Pages