Question: If {S1, S2, . . . , Sn} is a collection of convex subsets of a linear space X, any linear combination a1S1 + a2S2

If {S1, S2, . . . , Sn} is a collection of convex subsets of a linear space X, any linear combination a1S1 + a2S2 +............+ anSn, ai ∈ R is also a convex set.
Exercise 1.168 is a useful characterization of convex sets.

Step by Step Solution

★★★★★

3.51 Rating (168 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Combine Exercises 1164 and 1166 Exercises 1164 Assume ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

914-M-N-A-O (147).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

If {S1,S2,. . . , Sn} is a collection of cones in a linear space x, then ¢ Their intersection ¢ Their sum S1 + S2 + . . . + Sn are also cones in X.

Let {S1, S2, Sn} be a collection of subsets of a linear space X with S = S1 + S2 + ... + Sn. Let f be a linear functional on X. Then x* = x1* + x*2 + ... + x*n maximizes f over S if and only if x*i...

The collection of all convex subsets of a linear space ordered by inclusion forms a complete lattice.

If {S1, S2, . . . , Sn} is a collection of convex subsets of a linear space X, their sum S1 + S2 +...........+ Sn is also a convex set.

Let {S1, S2, . . . , Sn} be a collection of nonempty compact subsets of an m-dimensional linear space, and let x conv We consider the Cartesian product of the convex hulls of Si, namely Every point...

Extend exercise 3.210 to allow for noncompact Si, thus proving the Shapley-Folkman theorem. Exercise 3.210 Let {S1, S2, . . . , Sn} be a collection of nonempty compact subsets of an m-dimensional...

Let {S1, S2,..., Sn} be a collection of nonempty (possibly non convex) subsets of an m-dimensional linear space, and let Then 1. 2. where 3. with bij ¥ 0 and bij > 0 for at most m + n components....

The Basics of Financial Mathematics Spring 2003 Richard F. Bass Department of Mathematics University of Connecticut c These notes are 2003 by Richard Bass. They may be used for personal use or class...

(2) ( 30 points) We now show the VC dimension of linear threshold functions in n-dimensional Euclidean space is n+1, generalizing the result in Notes 05 . In the following, given a finite set...

Reconsider the experiment described in Problem 6-1. Suppose that the experimenter only performed the eight trials from replicate I. In addition, he ran four center points and obtained the following...

For the acetone absorber of Figure, assuming the use of sieve trays with a 10% hole area and &-in. holes with an 18-in, tray spacing, estimate: (a) The column diameter for a foaming factor of 0.85...

Why is it important for business units to continually revise their product or service offering? What new products or services is your employer, or an organization you are familiar with, considering?...

Explain the concept of Net Present Value (NPV) and its significance in capital budgeting decisions for a B.Com finance student.

As a project manager, you try to use empathic listening skills to help understand another persons frame of reference. In following this approach, you should a. Mimic the content of the message b....

The following data come from a table that was measured with high precision. Use the best numerical method (for this type of problem) to determine y at x = 3.5. Note that a polynomial will yield an...

Use Newton's interpolating polynomial to determine y at x = 3.5 to the best possible accuracy. Compute the finite divided differences as in Fig. 18.5 and order your points to attain optimal accuracy...

The data tabulated below were generated from an experiment initially containing pure ammonium cyanate (NH4OCN). It is known that such concentration changes can be modeled by the following equation: c...

Question 1 1 0 Points The CELLULOID CLOSET showed how movies have always showed positive portrayals of LGBT characters. True False Question 2 1 0 Points The CELLULOID CLOSET describes the Hays...

DVD Rentals The probabilities that a customer will rent 0, 1, 2, 3, or 4 DVDs on a single visit to the rental store are 0.25 , 0.25 , 0.1, 0.35, and 0.05 , respectively. Part: 0 / 2 Part 1 of 2...

Suppose a bond with no expiration date has a face value of $10,000 and annually pays a fixed amount of interest of $800. In the table provided below, calculate and enter either the interest rate that...