a) Prove that every convex set in Rn is connected. b) Show that the converse of part

Question:

a) Prove that every convex set in Rn is connected.
b) Show that the converse of part a) is false.
c) Suppose that f: R → R. Prove that f is convex (as a function) if and only if E: = {(x, y): y > f(x)} is convex (as a set in R2).

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

An Introduction to Analysis

ISBN: 978-0132296380

4th edition

Authors: William R. Wade

See More Books

Question Posted: Mar 29, 2016 04:21 AM

See More Questions

(a) Show that if a family of pdfs {f(x|): } has an MLR, then the corresponding family of cdfs is stochastically increasing in 6. (See the Miscellanea section.) (b) Show that the converse of part...
a) Prove that if k=1 ak converges, then its partial sums sn are bounded. b) Show that the converse of part a) is false. Namely, show that a series k=1 ak may have bounded partial sums and still...
a) Let [xn) and {yn} be sequences in X which converge to the same point. Prove that p(xn yn) 0 as n . b) Show that the converse of part a) is false.
Consider the two tables shown in Figure 6.37. The TerID column in the SALES_REP relation is a foreign key referring to the primary key column TerID in the TERRITORY relation. Figure 6.37 Relations...
Mansfield Corporation had 2013 sales of $100 million. The balance sheet items that vary directly with sales and the profit margin are as follows: Percent Cash ..............5% Accounts receivable...
In a small group, research a few collaboration platforms. Also share your own experience about the advantages and disadvantages of each. Which platforms might be best for which type of work or team?...
In Section 6.5 .2 Computation Lab: Normal Plots in Unreplicated Factorial Designs, the factorial effects were sorted based on their magnitudes before plotting the normal quantile plot shown in Figure...
A research hypothesis is that the variance of stopping distances of automobiles on wet pavement is substantially greater than the variance of stopping distances of automobiles on dry pavement. In the...
Discuss three social and legal concerns for the internet and e-business. Be specific and provide examples
Given the following data, calculate a level production plan, quarterly ending inventory, and average quarterly inventory. If inventory carrying costs are $6 per unit per quarter, what is the annual...
Suppose that V is open in R2, that H = [a, b] [0, c] V, that : V R is C2 on V, and that (x0, t0) > 0 for all (x0, t0) H. a) Show that, given > 0, there is a compact set K such that u(x, t) > -...
Suppose that f : R2 R is Cp on Î²r(x0, y0) for some r > 0. Prove that, given (x, y) Br(x0, y0), there is a point (c, d) on the line segment between (x0, y0) and (x, y) such that p-1
Does the equation for time dilation show dilation occurring for all speeds, whether slow or fast? Explain.
Which of the following is not associated with increased risk regarding insider cyber fraud? 1. Supervisors can implement strict controls over information system personnel. 2. Insiders are often aware...
Which of the following attributes of a fraud examination often provides the most compelling evidence of intent on the part of a fraudster? 1. Perceived pressure 2. The fraud act 3. Conversion 4....
Neural Net Evolution. A neural net typically starts out with random values for bias and weights; hence, it produces essentially random predictions when presented with its first case. What is the key...
Binary fission a. is a form of asexual reproduction. b. is a way for bacteria to reproduce c. begins with a single mother cell and produces two genetically identical daughter cells. d. All of the...
Describe the difference in roles assumed by the validation partition and the holdout partition.
Use Eulers method to approximate the indicated function value to 3 decimal places, using h = 0.1. dy dx xy + 4; y(0) = 0; find y(0.5)
In the figure, two loudspeakers, separated by a distance of d1 = 2.63 m, are in phase. Assume the amplitudes of the sound from the speakers are approximately the same at the position of a listener,...
An article in the American Industrial Hygiene Association Journal (Vol. 37, 1976, pp. 418422) describes a field test for detecting the presence of arsenic in urine samples. The test has been proposed...
An article in the Food Technology Journal (Vol. 10, 1956, pp. 3942) describes a study on the prospecting content of tomatoes during storage. Four storage times were selected, and samples from nine...
An experiment was conducted to investigate leaking current in a SOS MOSFETS device. The purpose of the experiment was to investigate how leakage current varies as the channel length changes. Four...
Excerpts from Andre Company's December 31, 2024 and 2023, financial statements are presented below: Accounts receivable Inventory Net sales Cost of goods sold Total assets Net income Total...
The York City Hospital has just acquired new equipment. The equipment cost $ 4 , 2 5 0 , 0 0 0 , and the organization spent $ 1 3 5 , 0 0 0 on upgrading the physical plant to the new equipment will...
Calculate the following showing all the necessary steps: Note: CPP rate to be used in the calculation is 5.95%. Note: Don't forget to deduct pay period exemption\ \ Tanya earns $25.00 per hour. This...

a) Prove that every convex set in Rn is connected. b) Show that the converse of part

Question:

Step by Step Answer:

a Suppose E is convex but not connected Then there is a pair of open sets U V which separates E Let ...View the full answer

An Introduction to Analysis

Students also viewed these Numerical Analysis questions