Convex Definition and 283 Threads

Here is the question: Here is a link to the question: Convex function ...? - Yahoo Answers I have posted a link there to this topic so the OP can find my response.

I'm looking for 1-2 rigorous books on convex analysis for someone who already has some exposure to convexity, linear and nonlinear programming in an applied course. It seems that Rockafellar (Convex Analysis) and Fenchel (Convex Cones, Sets and Functions) is the classic treatment. Is there a...

Hey! :o To show that a two-variable function is convex, we can use the hessiam matrix and the determinants. But the function is linear the matrix is the zero matrix. What can I do in this case?

This is not really a problem, more just a fact-based question. I have been using the Google, but I have been unable to find an answer. Is there a way to induce the answer that I am missing? :confused: Question: What range of distance does a double convex lens have to be from an object to...

In definition 2.17 of Rudin's text, he says that a set E is convex if for any two points x and y belonging to E, (1−t)x+ty belongs to E when 0<t<1. I learned that this means the point is between x and y. But I'm not able to see this intuitively. Can anyone help me "see" this?

What happens when you place an object into a two-lens system consisting of two convex lenses?

Homework Statement An object is 12.0 cm in front of a convex mirror. When the convex mirror is replaced with a flat mirror, the image distance is 5.0 cm farther away from the mirror. What is the focal length of the convex mirror? Homework Equations f=-1/2R 1/do+1/di=1/f The...

Convex function and convex set(#1 edited) Please answer #4, where I put my questions more specific. Thank you very much! The question is about convex function and convex set. Considering a constrained nonlinear programming (NLP) problem \[min \quad f({\bf x}) \quad {\bf x}\in \mathbb{R}^{n}...

Hi, just a few details prior: I'm trying to study techniques for maths proofs in general after having completed A level maths as I feel it will be of benefit later when actually doing more advanced maths/physics. With this question what is important is the proof is correct which means I don't...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Hello everyone, A friend of mine came up with this question in class and I really do not have a good answer. Suppose you have a convex lens that has been cut in half horizontally and the top half removed. The question is: Will the bottom half of the lens still form an image? I really...

Consider the quadratic function $\displaystyle q(\textbf{x}) = \frac{1}{2} \textbf{x}^T G \textbf{x} + \textbf{d}^T \textbf{x} + c$ on $\mathbb{R}^n$, where $\textbf{d}$ is a constant $n \times 1$ vector, $G$ is a constant $n \times n$ symmetric matrix and $c$ is a scalar. The gradient is...

1. A spherical mirror is used to form an image 5 times as tall as an object, on a screen positioned 5.0m from the mirror. a) Describe the type of mirror required b) Where should the mirror be positioned relative to the object 2. M = (Di)/(Do) & 1/(Di) + 1/(Do) = 1/(F) 3. I have...

I am stuck at the inequality proof of this convext set problem. $\Omega = \{ \textbf{x} \in \mathbb{R}^2 | x_1^2 - x_2 \leq 6 \}$ The set should be a convex set, meaning for $\textbf{x}, \textbf{y} \in \mathbb{R}^2$ and $\theta \in [0,1]$, $\theta \textbf{x} + (1-\theta)\textbf{y}$ also belong...

Homework Statement For my high school physics coursework I must investigate factors affecting the focal length of a lens. I have focused on radii of curvature and completed my data collection and verified the accuracy using the lens makers equation. However, in the conclusion I am really...

I was just wondering if a person wearing convex lenses looks at a light source... Will excess heat be generated in his eyes? As it is a converging lens.. Thanks!

Homework Statement So I am trying to understand this proof and at one point they state that an arbitrary compact subset of a Banach space, or a completely metrizable space is the subset of a finite set and an arbitrary convex neighborhood of 0. I've been looking around and can't find anything...

I don't understand what a convex hull is and what it does. does anyone think they can explain what exactly it is?

Edgar from Facebook writes: The sum of the measures of the interior angles of a convex polygon is ten times the sum of the measures of its exterior angles. Find the number of sides of a polygon. Hello could you please help me to solve this problem?

I'd like to know why a ray of light passing through the optical center of a convex lens does not get refracted at all. According to my knowledge, a ray of light will not get refracted if the angle of incidence is zero i.e. it is along the normal. With this in mind, I see how a ray traveling...

Homework Statement Show that a differentiable function f is convex if and only if the following inequality holds for each fixed point x0 in Rn: f(x) ≥ f(x0) + ∇tf(x0)(x-x0) for all x in Rn, where ∇tf(x0) is the gradient vector of f at x0. Homework Equations The Attempt at a...

While reading the reference Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html, there is something that I don't quite understand. Eq.(16.6) is an evolution equation for...

Homework Statement y= (x^2 -7) e^xThe Attempt at a Solution I'm trying to find inflection points by setting the second derivative=0 I found that the derivative is: ##2xe^{x}+x^{2}e^{x}-7e^{x}=0## ##e^{x}[2x+x^{2}-7]=0## Then, the 2nd derivative: ##e^{x}[(x-1)(x+5)]=0##, then the inflection...

Homework Statement An object is placed in front of a convex mirror whose radius of curvature is R. What is the greatest distance behind the mirror that an image can be formed? A. Infinity B. R C. R/2 D. No image can be formed. Homework Equations 1/do + 1/di = 1/f The...

Homework Statement Show that f(x) = x1x2 is a convex function on [a,ma]T where a \geq 0 and m \geq 1. Homework Equations By definition f is convex iff \forall x,y\in \Re \quad \wedge \quad \forall \lambda :\quad 0\le \lambda \le 1\quad \Rightarrow \quad f\left( \lambda x+(1-\lambda...

Homework Statement In a convex polygon of 6 sides, 2 diagonals are selected at random. The probability that they intersect in the interior of the polygon isHomework Equations The Attempt at a Solution There are 9 diagonals in a polygon of 6 sides. Therefore the total cases are 9C2. But how...

Homework Statement Prove that F = {x E R^n : Ax >/= b; x >/= 0} is a convex set. Yes x in non negative and A and b are any arbitrary Homework Equations The Attempt at a Solution Well I know A set T is convex if x1, x2 E T implies that px1+(1-p)x2 E T for all 0 <= p <= 1...

Homework Statement Using the definition of a convex set, show that the set in R^2 \{(x,y) \in R^2 \colon y \ge 1/x, x\ge 0\} Homework Equations An object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the...

Lens problem! Hi guys, When an object is placed in front of a convex lens ( not between Focus and optical centre ), the image formed on the other side should be inverted and real. My question is, with the mentioned placement of the object, will the image be laterally inverted? Further more...

I am having a problem understanding the mathematical reasoning of finding the description of image in convex mirror... We know the relation, 1/v + 1/u =1/f is true for all mirrors... we can prove this for both concave and convex mirrors... In case of convex mirrors we use the conventional...

The left end of a long glass rod 6.00cm in diameter has a convex hemispherical surface 3.00cm in radius. The refractive index of the glass is 1.60. Distances are measured from the vertex of the hemispherical surface. A.)Determine the position of the image if an object is placed in air on the...

So I know these equations 1/f = 1/p + 1/i m = -i/p f: focal length p: object distance from mirror i: image distance from mirror m: magnification Let's say that I have an object in front of a concave or convex mirror with the same |f|. p is much larger than the radius of curvature...

Hello all, I am reading a paper and there is one bit in the paper that I am having a bit of trouble understanding. Say V(a) is a convex function and then the paper has the following line: [V(-2pi + a] + V(2pi + a]] >= 2V(a) I am sure this relationship is simple and falls out somehow...

Existence of strictly convex functions with same "ordering" as convex one Consider any real-valued convex function c : R^n \rightarrow R. I am interested in whether there exists some strictly convex function d, that satisfies d(x) > d(y) if c(x) > c(y). That is, given a convex function, can we...

Homework Statement Givens: \forall x\ge 0:\quad f^{ \prime \prime }\left( x \right) \ge 0;\quad f\left( 0 \right) =0 Prove: \forall a,b\ge 0:\quad f\left( a+b \right) \ge f\left( a \right) +f\left( b \right) Homework Equations By definition, f is convex iff \forall x,y\in \Re \quad \wedge...

Hello everyone, I really run into a problem here. The magnification equation for mirrors describes such a relation: M=-distance of image/distance of object = height of image/height of object. (M=(-i/o)=h'/h). I understand how this formula can be proved using a ray diagram for concave...

I have a real-world situation I need help with before I run out and spend money. I have a wall that is in the shadows. I want to use a convex mirror to reflect sunlight onto the wall, covering as much of the surface area as possible. I will need a convex mirror with the right diameter and...

Let \alpha>0 and \gamma>0 and \beta>0 be real numbers. Let M={x∈R^{2}_{+} ∶\alphax_{1}+\gammax_{2}\leq\beta}. Prove M is a convex set. Prove that M is bounded. What does this set resemble (in economics)? I have a little idea of how to show that this set is convex, although, I know the condition...

Homework Statement Prove the Int<ABC is a convex set. Homework Equations The Attempt at a Solution 1. Int <ABC = H(A,BC) intersect H(C,AB) by the definition of interior. 2. H(A,BC) is convex and H(C,AB) is convex by Half-Plane Axioms I know I need to show the intersection of...

I need to prove the interior of <ABC is a convex set. I know it is. I started by defining the angle as the intersection of two half planes and using the fact that each half plane is convex. I am stuck on where to go from here.

let f(X) : Rn --> R be a function defined on convex set S s.t S is a subset of Rn (real space n-dim). Let f is positive throughout. Then define g(x) = (f(x))^2. Prove that if f(x) is convex then g(x) is also convex.

Homework Statement An object 2.00cm high is placed 12.0cm in front of a convex mirror with radius of curvature of 8.00cm. Homework Equations Where is the image formed? Draw a ray diagram The Attempt at a Solution do i use the equation m=-q/p? if so then (object distance)p= 12cm...

Prove that in any convex 2n-gon there is a diagonal not parallel to any side.

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...