Convex Definition and 303 Threads

Prove that in any convex hexagon there is a diagonal which which cuts off a triangle with area no more than one sixth of the area of the hexagon.

Let $X$ be a compact and convex subset of $\mathbb{R^2}$ Let $a^1, a^2 \in X$ such that $a^j = (a^j_1, a^j_2)$, $j=1,2$ Is $c= \sum_{i=1}^2 \mathbb{I}_{ i=j} a^j_i \in X \quad ?$

I am having trouble proving the following: Suppose that E is a convex region in the plane bounded by a curve C. Show that C has a tangent line except at a countable number of points. E is convex iff for every x, y \in E, and for every \lambda \in [0,1], (1-\lambda) x + \lambda y \in E...

Homework Statement Let K be the closed interval [0,1] and consider the function f(x)=x^2. Is f convex? Is f linear? help please :/ i don't even know how to set this up to check, our teacher didn't even get to this in class yet! Homework Equations The Attempt at a Solution

It seems like something that could (should?) be true, but with topology you never know (unless you prove it...). EDIT: I'll be more exact: let (X,\mathcal T) be a topological space with X a totally ordered set and \mathcal T the order topology. Say X is connected and A \subset X is convex (i.e...

Homework Statement It would make one of my proofs easy if it is true that " If E is an open connected set then it is convex''. I have been spending some time trying to prove this. Is this statement even true? Homework Equations Convex implies that if x is in E and y is in E then εx+(1-ε)y...

Homework Statement Let S C Rd be open and convex. Let f be C1(S). Prove that if f is strictly convex, then f(y) > f(x) + grad f(x) o (y-x) for all x,y in S such that x≠y. (note: "o" means dot product) Homework Equations Strictly convex functions The Attempt at a Solution Suppose f...

Homework Statement Show that it is possible to cut any convex polygon into 4 pieces of equal areas by using two cuts perpendicular to each other. Homework Equations None, it's just a proof I found on the back of my book. The relevant chapter is Continuity, the maximum principle, and...

Theorems about convex functions often look like the following: Let f: S->R where S is a convex set. Suppose f is a convex function... So here are my questions: 1) For a convex function, why do we always need the domain to be convex set in the first place? 2) Can a convex function be...

From what I recall, determining the focal length of a double convex lens just involves an infinite light source and holding the lens close to an image-capturing surface such as a wall and adjusting the distance until the image is at its sharpest...the distance from the lens to the surface is the...

I am trying to find the focal length of a convex lens, so i let the sun shine through my window to create an image on my wall. Is that image the focal point or is it just the distance of the image? And if it is the distance of the image, how do I put it into the thin lens equation, since the...

I've been trying to use Taylor's theorem with h = (y-x)/2 to show that a twice differentiable function for which the second derivative is positive is midpoint convex (ie, f( (1/2)*(x+y) ) \leq (1/2) * (f(x)+f(y)) ). (It's not a homework problem.) The problem I end up with this is that I'm not...

Dear Friends, I am trying to to find out if it is possible for the convex spherical lens to focus an ultraviot light to a single spot, and what is the power of the lens? Thank you very much for your help

I am interested in comparing the effectiveness of GRIN lenses vs aspheric-convex ones for collimating laser beams. I can imagine some practical advantages however I would be interested in any information comparing the advantage of one method over the other from an optical standpoint. Thanks...

Hi, I hope someone can help me out with this problem: Let set S be defined by (x in En :f(x) <=c} f: En -> E1 is convex and differentiable and gradient of f(xo) is not 0 when f(x) = c. Let xo be a point such that f(xo) = c and let d = gradient at xo. Let lamda be any positive number and...

\frac{d}{dx}f(x)=\frac{d}{dx}[ \frac{1}{2}x_{}^{T}Qx-b_{}^{T}x] how to get this derivative, what is the answer? is there textbook describe it?

\frac{d}{dx}f(x)=\frac{d}{dx}[ \frac{1}{2}x_{}^{T}Qx-b_{}^{T}x] how to get this derivative, what is the answer? is there textbook describe it?

1. Let A and B be convex cones in a real vector space V. Show that A\bigcapB and A + B are also convex cones.

Homework Statement I have to determine if [e][/x] is a convex function. If it is then show proof. I know its a convex function by looking at the graph, Iam stuck at prooving it mathematically though. Homework Equations The function is f(x)=e^x. The Attempt at a Solution I am...

Homework Statement Prove that every convex polytope is convex and closed. Homework Equations C=\{ \sum_{j=1}^n x_j a^j | x_j \geq 0, \sum_{j=1}^n x_j = 1\} is a convex polytope The Attempt at a Solution I've already proven the convexity portion. To prove C is closed, I let \{ b^N...

Let \mathcal B(\Omega) be the Borel algebra of \Omega (the σ-algebra of Borel sets in \Omega). I understand that if we define a "convex combination" of probability measures by \bigg(\sum_{k=1}^n w_k\mu_k\bigg)(E)=\sum_{k=1}^n w_k\mu_k(E), then every convex combination of probability measures is...

Homework Statement difference between convex lens formula and convex mirror formula The Attempt at a Solution for convex mirror you make focal point negative and use the 1/f = 1/do + 1/di but for convex lens do you do the same? you don't make the f negative i think am i...

Homework Statement I did an experiment with a convex lens, object, and image. As the lens moved farther and farther away from the object, the image decreased its distance (to remain in focus) from the object as well up to a certain point. After that point, as the lens increased its distance...

I know that for concave lenses, the focal length is one half the length of the radius of the circle. Is this also true for convex lenses? Thanks

I've been doing some calculations on lenses, and I'm a little confused about some of the diagrams I've seen. Looking at a ray diagrams, I see the light passing through the lens, at one angle, and then converging on a focal point from there. Using snells law to calculate refraction through the...

I am trying to ultimately find the projector onto a convex set defined in a non-explicit way, for a seismic processing application. The signals in question are members of some Hilbert Space H and the set membership requires that they must correlate with each other above some scalar \rho, given...

Homework Statement A body is positioned 40 cm from the left of a lens (f=30) and the lens is positioned 100cm left of a convex mirror (|R|=60), where will be the image and what type and magnitude will it have? Homework Equations 1/u + 1/v = 1/f The Attempt at a Solution I tried to...

Given a Normed Linear Space, prove that all open balls are convex. A, a subset of the space is said to be convex if, for all pairs of points (x,y) in a, the point z = x + t(y-x) belongs to A. (t goes from 0 to 1).

Hello :) I have been giving a mathematical problem. But I find difficulties solving this. Therefore, I will be very grateful if anybody might wanted to help? The problem is "Let K be a compact convex set in R^n and C a closed convex cone in R^n. Show that ccone (K + C) = C." - Julie.

Homework Statement A 1.5 cm high diamond ring is placed 20 cm in front of a convex lens whose radius of curvature is 30 cm. a) What is the position and the size of the image? b) What magnification does this lens have? Homework Equations 1/f= 1/di + 1/do m= -di/do or hi/ho The Attempt at a...

Homework Statement Following data recorded: Di (distance image to lens) 6.0 8.0 10.0 12.0 14.0 Magnification 0.2 0.6 1.0 1.4 1.8 Graph magnification against di and use the graph to find the focal length of the lens. Homework Equations...

When an object is placed at the focal point of a convex lens, the image is formed at infinity, yet it is called a highly magnified image. The fact being that after passing through the convex lens the rays run parallel or rather as a beam. My question is that if the image is formed at...

The problem: "A convex spherical mirror is 25 ft from the door of a convenience store. The clerk needs to see a 6 ft. person entering the store at least 3 inches tall in the mirror to identify them. What is the radius of the mirror?" d_obj = do = 25 ft = 300 inches h_img = hi = 3 inches...

Homework Statement A child holds a candy bar 16.5 cm in front of the convex side-view mirror of an automobile. The image height is reduced by one-half. What is the radius of curvature of the mirror? 1Your answer is incorrect. cm Homework Equations 1/f = 1/do + 1/di...

So this is the case - I've bought some shocking laser tag guns to play with my friends, though I am unsatisfied with their range which is merely 5 meters. I was wondering if I could add some convex lenses in front of the laser gun in order to increase the range of the laser, would that work...

Homework Statement Hello! I'm having some trouble trying to understand basic concepts of Convex Analysis (I study it independently). In particular, I have a book (Convex Analysis and Optimization - Bertsekas) which gives a definition for the convergence of a sequence of sets: Homework...

Homework Statement I am trying to understand the reasoning behind the following statement from Boyd's Convex Optimization textbook, page 50, line 9-11: "f must be negative on C; for if f were zero at a point of C then f would take on positive values near the point, which is a contradiction."...

Homework Statement Suppose that f is analytic on a convex set omega and that f never vanishes on omega. Prove that f(z)=e^(g(z)) for some analytic function g defined on omega. Hint: does f'/f have a primitive on omega? Homework Equations f(z)=\sum_{k=0}^\infty a_k(z-p)^k The...

Homework Statement Find focal length for meniscus convex (3) Given r = 15, and n = 1.5 Homework Equations Lens maker's equation The Attempt at a Solution I can find the rest except #3 and #6 (they are opposite sign of each other) I had (1/r1 - 1/r2) gives zero... which is wrong... I...

Homework Statement Two thin convex lenses (1 and 2) and a small object are arranged as shown. a) Use the three principal rays to determine the location of the image of the object produced by lens 1. . Object --------.F1---0(Thin lens 1)---.F1-----------.F2----0(Thin lens 2 -...

Homework Statement A small bulb is placed in front of a convex lens. a) Suppose that the bulb is placed as shown. Using all three principal rays, draw an accurate ray diagram to determine the location of the image. Label the image location...

I am trying to find the convex hull of a finite set in a hyperbolic space, particularly the Poincare disk, but the Upper Half plane works as well. I know the following equivalent definitions of the Convex Hull: 1) It is the smallest convex set containing the points 2) If the set is...

Homework Statement A double convex lens has a diameter of 5 cm and zero thickness at its edges. A point object on an axis through the center of the lens produces a real image on the opposite side. Both object and image distances are 30 cm, measured from a plane bisecting the lens. The lens has...

Homework Statement Prove that the intersection of a number of finite convex sets is also a convex set Homework Equations I have a set is convex if there exists x, y in the convex S then f(ax + (1-a)y< af(x) + (1-a)y where 0<a<1The Attempt at a Solution i can prove that f(ax + (1-a)y) <...

Homework Statement W(\theta - \delta) the loss function. \theta the true parameter. \delta an estimator of \theta W a smooth, non-negative, symmetric, convex function. p(\theta | x) the posterior density of the parameter \theta. Prove that, for normal posterior density p(\theta | x)...

I will be thankful if the following points are clarified. 1. While deriving the lens formula for a convex lens-say, 1/f= 1/v - 1/u where u and v are the object and image distances, the minus sign is obtained after applying sign conventions. But, I don't understand the logic behind taking the...

Homework Statement In my textbook, the author briefly makes a statement that affine functions are both concave and convex, how is that true? and how can it be proven? Homework Equations The Attempt at a Solution

Homework Statement Consider a feasible region S defined by a set of linear constraints S = {x:Ax<b} Prove that S is convex Homework Equations All what i know is that, a set is convex if and only if the elements x, and y of S ax + (1-a)y belongs to S for all 0 <a < 1 The...

Homework Statement In the given arrangement pulley P1 and P2 are moving with constant speed vo downward and the centre of the pulley P lies on the principal axis of a convex mirror having radius of curvature R. Find the speed of image of pulley P when it is at a distance x from the surface...

Homework Statement A dentist uses a curved mirror to view teeth on the upper side of the mouth. Suppose she wants an erect image with a magnification of 2.00 when the mirror is 1.25 cm from a tooth. (Treat this problem as though the object and image lie along a straight line.) What kind of...