Problem regarding the convex function

Assignment Help Basic Statistics

Reference no: EM13990801

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: EM13990801

Questions Cloud

Unique nonnegative radon measure : Recall that a Radon measure on R is a function µ into R, defined on all bounded Borel sets and countably additive on the Borel subsets of any fixed bounded interval.

Project breakdown risks and staffing : Project Breakdown, Risks and Staffing - PMI is one of the world's largest not-for-profit membership associations for the project management profession

How can global city accommodate both global & local identity : Explain two or three consequences that may occur if a global city exiled the middle and working class?

Convex function defined on a convex : 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.

Problem regarding the convex function : 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 funct..

Case study about should films shown in the uae be censored : Write an ethical case study about "Should films shown in the UAE be censored?" in ethics standards not opinion based or law based.

Find the magnetic field on the z axis : The figure below shows two long, straight wires in the xy plane and parallel to the x axis. One wire is at y = -6.0 cm and the other wire is at y = +6.0 cm. The current in each wire is 22 A. Find the magnetic field on the z axis at z = +7.8 cm if ..

Problem regarding the open interval : 1. Suppose that a real-valued function f on an open interval J in R has a second derivative f II on J . Show that f is convex if and only if f II ≥ 0 everywhere on J .

Purpose of government as envisioned & articulated : In particular, assess whether, in your view, its continued funding is critical to the country in light of the federal debt, redundancies, free market alternatives &, especially whether its actual existence is consistent with the purpose of governm..

User Account

All Pages