Convex Definition and 303 Threads

I am taking a self-study individual course on convex analysis but I'm having some troubles with the basics as I'm trying to do the exercises in my notes. I'm asked to consider the space C[0,1] of continuous, complex-valued functions on [0,1], equipped with the supremum norm \|\cdot\|_{\infty}...

For the equation: contact stress = {(1 / (pi[((1-v1^2)/E1)) + ((1-v2^2)/E2))) ^ 0.5} * {((Fn/b) * (Sum (1/pi)))^0.5} Where Sum (1/pi) = [(1/p1) - (1/p2)] for concave shapes in contact with convex shapes Sum (1/pi) approaches 0 as the two radii get closer, however when the two radii...

hi :) if someone have any idea ? what is the difference between a convex norm and strong convex norme ?

A convex lens of focal point 0.8m focuses a real image of the moon onto a screen. If the moon subtends an angle of 0.5 degrees to an observer on earth, calculate the size of the moons image on the screen? Thats the question guys, I am really struggling to find the appropriate equation as we...

1)Why when a convex lens having radius of curvature R, focal length f and refractive index μ is bisected horizontally along the principal axis its focal length remains the same, whereas when it is bisected vertically focal length becomes 2f? Does the same thing happens in the case of...

Hi, It is well known by Varignon's Theorem and the Triangle Inequality that in a convex quadrilateral ABCD with midpoints a of side AB, b of side BC, ... d of side DA and perimeter p, \[ AC + BD < p\] where AC and BD are the diagonals of the quadrilateral. However, how do I obtain...

Hi, It is well known by Varignon's Theorem and the Triangle Inequality that in a convex quadrilateral ABCD with midpoints a of side AB, b of side BC, ... d of side DA and perimeter p, AC + BD < p where $AC$ and $BD$ are the diagonals of the quadrilateral. However, how do I obtain...

Hi, I am presently writing Python code to solve the following problem; fit the largest volume polytope (P1), with a given number of faces, into a known other polytope (P2) (with more faces than the one being fitted inside). Both polytopes are convex and are described by the feasible region of...

Why is a telephoto lens considered to have a longer focal length than that of single convex lens?

Homework Statement A convex mirror of focal length 10cm forms an image that is one quarter the size of the object. Find the position of the object and the image. Homework Equations -1/f = 1/u + 1/v m = v/u The Attempt at a Solution m = 1/4 f = 10cm -1/10 = 1/u + 1/0.25u I think...

Homework Statement Hi there, I have a set similar to this \{(x,y)\in{\mathbb{R}^2}:x^2+y^2\neq{k^2},k\in{\mathbb{Z}\} (its the same kind, but with elipses). And I don't know if it is convex or not. If I make the "line proof", then I should say no. What you say? Bye there, and thanks.

Homework Statement 1. A small bulb is placed in front of a convex lens. It is placed at one focal point of the lens. Draw at least five rays from the bulb that pass through the lens. Where is the lens located in this case? Explain. (Hint: How are the rays that have passed through the...

f:(a,\infty)->R i want to prove, that, if function is convex, then: if exist x_1 \in R, exist x_2>x_1 : f(x_2)>f(x_1) then: for all x_3>x_2 for allx_4>x_3 : f(x_4)\ge f(x_3)\ge f(x_2) in other words: convex function is either decreasing on whole domain, or it starts to increase from some point...

Homework Statement Show that the closed unit ball {x E V:||x||≤1} of a normed vector space, (V,||.||), is convex, meaning that if ||x||≤1 and ||y||≤1, then every point on the line segment between x and y has norm at most 1. (hint: describe the line segment algebraically in terms of x and y...

Homework Statement An object is placed 32.7 cm from the reflecting surface of a convex mirror having a focal length of -10.8 cm. Determine the image distance. Homework Equations IDK The Attempt at a Solution Could someone please tell me what equation I should use? Thanks!

Homework Statement The virtual image produced by a convex mirror is one-quarter the size of the object. b) What is the focal length of this mirror? Homework Equations 1/di = 1/f - 1/do The Attempt at a Solution i found di to be -9.3 cm but when i solved for f i keep getting it...

Homework Statement A convex spherical mirror has a radius of curvature R = 20.0 cm and produces an upright image precisely one-quarter the size of an object. Calculate the separation distance between the object and its image? Homework Equations M = (image height)/(object height)...

Homework Statement Let f:\mathcal{O}\subset\mathbb{R}^n\rightarrow\mathbb{R}, \mathcal{O} is an open convex set. Assume that D^2f(x) is positive semi-definite \forall x\in\mathcal{O}. Such f are said to be convex functions.Homework Equations Prove that f((1-t)a+tb)\leq...

I want to know when a vehicle travels at a constant velocity along a convex and concave bridges, in what bridge the normal reaction of the vehicle is higher... I think it is on cancave bridge... am I right? because I got it by equations...

Hey, When dealing with convex mirrors I know that the image will always be virtual,upright and diminished. So when I am using descartes's formulae i know that for convex mirrors that the focal length is negative, but does my 1/di become negative? because virtual images have a negative height...

Homework Statement For each thin lens shown in the figure , calculate the location of the image of an object that is 18.5 {\rm cm} to the left of the lens. The lens material has a refractive index of 1.50, and the radii of curvature shown are only the magnitudes...

Homework Statement At an intersection of hospital hallways, a convex mirror is mounted high on a wall to help people avoid collisions. The mirror has a radius of curvature of 0.510 m. -What is the image distance for a patient 11.9 m from the mirror? (Use the correct sign conventions.)...

Homework Statement Let C be a closed convex subset of Rn. If y is not in C, show x* in C is the closest vector to y in C if and only if (x-y)(x*-y) >= ||x* - y||2 for all x in C. Homework Equations Theorem. Suppose that C is a convex set in Rn and that y is a vextor in Rn that is not in C...

Homework Statement How do I show that if f\in C^2 \text{(}\mathbb{R}\text{)} is convex then the function yf(y^{-1}\textbf{x}) is convex on (x,y):y>0?Homework Equations I know the standard definitions and whatnot about convexity, but I tried chugging through the algebra and didn't have any...

Homework Statement Suppose that no three of the diagonal of a convex n-gon meet at the same point inside of a n-gon. Find the number of different triangles the sides of which are made up of the sides of the n-gon, the diagonals and segments of the diagonals. How to find it~@~? Homework...

Homework Statement Hello Two plano convex with f1=f2=30mm and R1=15.45mm and R2=-R1. The two plano convex have the same thickness too (3.16 mm) and their curves colide in the propagation axis. I believe you understand what i mean What is the focal length of the two plano convex system...

Homework Statement Show that the union of convex sets does not have to be convex. Homework Equations The Attempt at a Solution Is it enough to just show a counterexample? Or is that not considered a complete proof? My example is...S = {1} and T = {2}.

Homework Statement What is the image distance for an automobile 4.5 m in front of a convex miror with a 0.15 m focal length? I found this to be 0.155 m The question I'm having a problem with is: What is the magnification? Homework Equations magnification = - (image distance/object distance)...

1. Shiny lawn spheres placed on pedestals are convex mirrors. One such sphere has a diameter of 40 cm. A 12 cm robin sits in a tree 1.5 m from the sphere. a) Where is the image of the robin? b) How long is the robin's image? 2. 1/f = 1/do + 1/di 3. a. 1/f = 1/do + 1/di...

Convex mirror problem help?? Homework Statement a convex mirror has a focal length of 75cm. an object with a height of 2.0m is 6.0m from the mirror. a) what is the distance from the object's virtual image to the mirror? b) what is the height of the object's image? Homework Equations...

Homework Statement Prove that the virtual image in a convex mirror is always smaller than the real object. Homework Equations m = -\frac{d_{i}}{d_{O}} The Attempt at a Solution Not a homework problem. Something which is bothering me, and haven't been able to prove yet. Thanks!

Homework Statement An object is located 14 cm in front of a convex mirror, the image being 7 cm behind the mirror. A second object, twice as tall as the first one, is placed in front of the mirror, but at a different location. The image of this second object has the same height as the other...

Hi everyone, I've got this problem to solve: My problem is that I don't fully understand the question. I have found such definition of convex hull: So I do have to prove, that all the roots of W'(z) [let's denote them as z'_k] must be able to be written in such form: z'_k = \sum_{k=1}^n...

Homework Statement Given only the distance between the object and its image (p +q), how is it possible to find p, the distance of the object from the convex lens? Homework Equations 1/p + 1/q = 1/f in this case, p + q =40 also, the image is smaller than the original, so p > 2f The...

I had a quick question: Is the following proof of the theorem below correct? Theorem: If C is a convex subset of a Topological vector space X, and the origin 0 in X is contained in C, then the set tC is a subset of C for each 0<=t<=1. Proof: Since C is convex, then t*x + (1-t)*y...

Homework Statement I solved the problem, but upon further observation, I discovered that what I did didn't make sense. (these are rounded numbers, which shouldn't make a difference in my question) do = 13 cm (do is what I solved for in the problem, and according to the online system it was...

Homework Statement The question is "Show that in the case of any linear program, every convex combination of optimal extreme points is optimal." Homework Equations ok so if (x_1,...,x_n) is a list of the optimal points then a_1(x_1)+ ...+a_n(x_n) is the convex combination st a_i>0...

Hi all, I want to know whether it is correct that every convex polygon has an inner-circle (& hence an inner-radius). I think it is only possible for triangle and for regular polygon. Am I right? If there is any convex N-gon having sides a_1,a_2,...,a_N which has an incircle, then...

Homework Statement Let x* be an element of a convex set S. Show that x* is an extreme point of S if and only if the set S\{x*} is a convex set. Homework Equations (1-λ)x1 + λx2 exists in the convex set The Attempt at a Solution I'm not too sure what S\{x*}, I asssumed it was...

Homework Statement What is the number of edges of a convex polytope with n vertices all of whose faces are triangles. Homework Equations # of faces + #of vertecies = # of edges + 2 The Attempt at a Solution My reasoning is as follows: n/3 + n = # of edges + 2 4n/3 - 2 =...

Does anyone have a textbook on multivariable complex analysis, and do they define a holomorphically convex domain in terms of the union of an ascending series of compact subsets? If so, how exactly does the definition go? Was it D=\bigcup K_n where K_n is holomorphically convex and compact...

How do i prove this? Let S = {(X1, …., Xn) € R^n | Xi ≥0, X1 + … +Xn = 1}. Show that S is convex. Suppose f(Si) = {X1, X2,..., Xn} AND g(Si) = {X1 + X2+ ...+Xn} but If do that, then f(Si) and g(Si), both will increase. Now I'm not sure where to go from here. *PS: Both functions are...

How do i go about showing that if f(x)=\left\|x\left\| then f(x) is a convex function. I'm thinking in the direction of the triangle inequality but don't know how to go about it. Any clues? thanks

Homework Statement Let a function f : R => R be convex. Show that f is necessarily continuous. Hence, there can be no convex functions that are not also continuous. Homework Equations The Attempt at a Solution F is continuos if there exist \epsilon >0 and \delta>0 such that |x-y|<...

Homework Statement Assume f is a continuous real function defined in (a,b) such that f(\frac{x+y}{2})<=\frac{f(x)+f(y)}{2} for all x,y in(a,b) then f is convex. Homework Equations The Attempt at a Solution my attempt is to suppose there are 3 points p<r<q such that f(r)>g(r)...

In a conversing lens, if the object is not within the focal point, then the magnification is "-". But in the case where the object is placed in the focal point, then the magnification is "+"?

Homework Statement A converging lens (convex lens) creates an image that is 2 times the size of the object. The object is placed: Homework Equations -di/do = hi/ho 1/f = 1/di + 1/do The Attempt at a Solution I thought the answer was that the object is place two focal lengths...

Hello! I'm trying to find some information about my problem but it doesn't seem very easy. 1 - I have a convex polyhedron defined as the intersection of several half-planes. 2- Now I would like to obtain a triangularization of the polyhedron surface in the best way. Can anyone indicate me...

Can anyone explain to me the procedure that one would follow to you find the focal length of a concave lens using a convex lens? Thanks.

Hi all, if light was to go straight through a convex lens would it come out straight or inverted to a point? also if light was to go through the same lens but goes through going inwards how would the light be refracted?