In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points, it contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty).
For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.
The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset A of Euclidean space is called the convex hull of A. It is the smallest convex set containing A.
A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis.
The notion of a convex set can be generalized as described below.
Hi,
I have problem to prove that the following inequality holds
I thought of the following, since it is a convex function and ##x_1 < x_2 <x_3## applies, I started from the following inequality ##f(x_2) \leq f(x_3)## and transformed it further
$$f(x_2) \leq f(x_3)$$
$$f(x_2)-f(x_1) \leq...
Since the lens is convex, I figured that the points where the red and blue light focus on the optical axis would be equal to their respective focal lengths (f), given that the incoming rays are parallel to each other and perpendicular to the lens.
Solving this got me to 1/fred = (1.52 -...
You can see from the picture that the teacher has circled the arrows which shows light coming from image to the eye, and drew it in the opposite direction saying the light goes from the eye to the image.
The marking scheme of this paper only says the correct direction does not specify which is...
but for me there is no solution to this inequality...
I never wait for a ready answer but I've already spent 4 hours on it and I still don't know what to mark...
Hello everyone,
I am struggling to get insight into a certain set in 4D space. Given is a closed path in 4D-space with constant Euclidean norm
$$\vec{\gamma} (\theta):[0,2\pi]\to\mathbb{R}^4, \ \ \vec{\gamma}(0)=\vec{\gamma}(2\pi), \ \ ||\vec{\gamma}(\theta)||_2 = \mathrm{const.}$$
I am looking...
f(x)=ln(|x1|+1)+(-2x1 2 +3x2 2 + 2x3 3) + sin(x1 + x2 + x3), for this problem in particular would be it be sufficient to find the Hessian and to see if that matrix is semi positive definite to determine if it convex?
Hello,
I am currently working on the proof of Minkowski's convex body theorem. The statement of the corollary here is the following:
Now in the proof the following is done:
My questions are as follows: First, why does the equality ##vol(S/2) = 2^{-m} vol(S)## hold here and second what...
The Korean textbook standard defines the convexity of the function as an open section. Many textbooks and university calculus textbooks define the convexity of the curve as an open section. However, some textbooks define convexity as closed sections.
Do you think it is right to define the...
Hi,
I was working through the following problem and I am getting confused with the solution's definition of the dual.
Problem:
Given the optimization problem:
minimize ## x^2 + 1 ##
s.t. ## (x - 2) (x - 4) \leq 0 ##
Attempt:
I can define the Lagrangian as:
L(x, \lambda) = (x^2 + 1) + \lambda...
Let ##S## be a set of n geometric objects in the plane. The intersection graph of ##S## is a
graph on ##n## vertices that correspond to the objects in ##S##. Two vertices are connected
by an edge if and only if the corresponding objects intersect.
Show that the number of intersection graphs of...
I wanted to know about spherical aberration in a biconvex and plano convex lens as I was planning an experiment with them.
I was reading about them and came upon the following passage.
I don't know whether the given equation is an empirical one or a derived equation.
Can anyone help me if you...
Hello-
I am starting a project to get direct sunlight onto the house by placing convex mirrors on the outside stone fence of the backyard since it is the only place of my property that is not shaded by other houses in the neighborhood. For that, I need to calculate the size of the mirrors...
Hey!
A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex if for all $x,y\in \mathbb{R}^n$ the inequality $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$ holds for all $t\in [0,1]$.
Show that a twice continuously differentiable funtion $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex iff the...
I try to proof it but i got stuck right here, i want your opinions
Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?
Summary:: Be the set X of vectors {x1,...,xn} belong to the vector space E. If this set X is convex, prove that all the convex combination of X yet belong to X. Where convex combination are the expression t1*x1 + t2*x2 + ... + tn*xn where t1,...,tn >= 0 and t1 + ... + tn = 1
I tried to suppose...
I am reading A Course in Mathematical Analysis Volume 1 by D. J. H. Garling, and I am having trouble in the following demonstration of Section 2 Differentiation. part 4 of the test, the first part of the second inequality does not make sense, I do not understand its justification. I hoped they...
When you put a convex and then concave lens in front of a light source, the light will be parallel but narrower than when it came in such as in a laser beam expander/compressor. Using a pen laser and a convex and concave lens, is it possible to focus the beam by putting a convex then a concave...
This exercise is located in the vector space chapter of my book that's why I am posting it here.
Recently started with this kind of exercise, proof like exercises and I am a little bit lost
Proof that given a, b, c real numbers, the set X = {(x, y) E R^2; ax + by <= c} ´is convex at R^2
the...
Hello everyone this was a problem on one of the exams from last year and I'm having trouble with the last point ##3##
my solution for ##1##
$$\frac{1}{2\pi i}\int_{|z|=r}\frac{f(z)}{z}(1+\frac{z}{2re^{i\theta}}+\frac{re^{i\theta}}{2z})dz =$$
I divided this integral into 3 different ones and...
Here is this week's POTW:
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Suppose $f : (a,b) \to \Bbb R$ is a convex function. Show that $f$ is differentiable at all but countably many points and the derivative is nondecreasing.
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Remember to read the...
Summary: Hi, I'm doing an assessment for Physics on Optics topics, but I can't really explain how the CV affect the RV
CV : Focal length of convex lens
RV : Temperature of water after 20 minutes under the sun
How I should explain for the temperature difference? 🤔
Please help, Thanks.
Hi PF!
Do you know what a strictly convex function is? I understand this notion in the concept of norms, where in the plane I've sketched the ##L_1,L_2,L_\infty## norms, where clearly ##L_1,L_\infty## are not strictly convex and ##L_2## is. Intuitively it would make sense that any...
Homework Statement
A strip of width w is a part of the plane bounded by two parallel lines at distance w. The width of a set ##X \subseteq \mathbb{R}^2## is the smallest width of a strip containing ##X##. Prove that a compact convex set of width ##1## contains a segment of length ##1## in every...
Hi, initially I am aware of the fact that when slater condition holds, then dual optimum equals primal optimum in convex optimization. But if slater condition does not hold then dual gap exist. When we have nonlinear nonconvex optimization we apply convexification of constraints including...
Hello! (Wave)
We are given a continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f \left( \frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}, \forall x, y \in \mathbb{R}$. I want to show that $f$ is convex.I have tried the following:
Let $\lambda \in [0,1]$.
We have that $f(\lambda...
Hello! (Wave)
I want to find two convex functions $f,g: \mathbb{R} \to \mathbb{R}$ such that $f(x)=g(x)$ iff $x$ is an integer.I have thought of the following two functions $f(x)=e^x$, $g(x)=1$.
Then at the $\Rightarrow$ direction, we would have $f(x)=g(x) \Rightarrow e^x=1 \Rightarrow x=0 \in...
Homework Statement
A thin convex lens of focal length 20 cm is split into two parts along the principal axis, now there is a 2mm gap between the top and bottom parts, an object O is placed at 10 cm from the lens, the distance between the two virtual images will be?
Homework Equations
1/v - 1/u...
Hello
My question has two parts:
1) Why is plano convex lens used in slide projector? Why can't we use simple convex lens in it?
2) Why are two plano convex lenses used in slide projector? Why can't we use only one?Thanks!
If you would allow me to ask...
if i have two convex functions , and i was to place one inside the other, i.e. convolute them...what could be said in general about the resultant function.
what information about the original functions can be taken from the positions of the minima.
and is there...
The original problem for anyone that can read Chinese:
https://zerojudge.tw/ShowProblem?problemid=b221
The problem defines a convex polygon with multiple points located in the first quadrant and the required task is to find a linear function y = ax that can spilt the polygon into two parts each...
Homework Statement
Let ##C \subset \mathbb{R}^n## a convex set. If ##x \in \mathbb{R}^n## and ##\overline{x} \in C## are points that satisfy ##|x-\overline{x}|=d(x,C)##, proves that ##\langle x-\overline{x},y-\overline{x} \rangle \leq 0## for all ##y \in C##.
Homework Equations
By definition...
Homework Statement
[/B]
Let X be a vector space over ##\mathbb{R}## and ## f: X \rightarrow \mathbb{R} ## be a convex function and ##g: X \rightarrow \mathbb{R}## be a concave function. Show: The set {##x \in X: f(x) \leq g(x)##} is convex.
Homework Equations
[/B]
If f is convex...
Hello everyone,
In stand up paddle boarding, the correct way to use the paddle when moving forward is to have the paddle convex side facing rearward while the paddle pushes water backward. This is explained by the fact that the paddle generates a lift force pointing backward if the convex side...
Homework Statement
Two slits (of width ##a=39 \mu m##) are lighted up with a monocromatic wave of ##\lambda=632,8 nm##. The distance between slits and the screen id ##D=4 m##. The distance between the slits id ##d=195 \mu m##.
In front of the slits there are a convergent lens with focal length...
Homework Statement
Homework EquationsThe Attempt at a Solution
The photoelectric current is directly proportional to the intensity of the light falling on it .
It will not depend on the focal length of the lens .
When the lens of half the diameter is used , intensity is halved . This...
When does a convex lens behhave as a simple microscope and what are the conditions for the object and the lens to act if there are any?Is the magnifying power of microscope fixed beyond which it will act as a regular convex lens
Homework Statement
I'm supposed to make a ray diagram of two convex lenses when light initially hits the first lens at parallel rays. When f(total) goes to infinity, the distance between the two lenses = the sum of the focal lengths of each lens.
Homework Equations
The Attempt at a Solution...
Homework Statement
Prove that the conjugate of ##g(x) = f(Ax + b)## is ## g^*(y) = f^*(A^{-T}y) - b^TA^{-T}y ## where A is nonsingular nXm matrix in R, and b is in ##R^n##.
Homework Equations
This is from chapter 3 of Boyd's Convex Optimization.
1. The conjugate function is defined as ##...
Homework Statement
I have noticed that in cars in India on rearview mirror(convex) it is written that "objects in mirror are closer than they appear".
But for convex mirror of focal length 1 metre and object distance 39 metre, the image distance is 39/40 metre .
Which tells that image is closer...
Homework Statement
Derive the mirror equation for a convex mirror (Si*So=f^2). Do not use a Ray heading toward the center of curvature point (C).
Si - distance between the object's image and the focal point.
So- distance between the object and the focal point.
Do- distance between the mirror...
Homework Statement
A convex spherical mirror with a focal length of magnitude 24.0 cm is placed 22.0 cm to the left of a plane mirror. An object 0.300 cm tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple...
Does the orientation of a plan convex lens affect it's focal lenght?
If I put il like in the first example in the photo and makeep the calculation using the lens maker equation I get f positive
What happens if I turn the curved face on the right?
Using the lens maker equation now the radius...
Hey! :o
I want to show that the function $f(x)=(1+x)^n, x\geq -1$ is for $n\in \mathbb{N}$ convex.
So that the function is convex it must hold $f''(x)>0$.
The second derivative is $f''(x)=n(n-1)(1+x)^{n-2}$.
It holds that $n>0$ and $n-1\geq 0$. We also have that $x\geq -1$. Therefore, we...
Homework Statement
Homework EquationsThe Attempt at a Solution
Consider S = {(1,1)} and T = {(0,0)}
Clearly, S and T is convex
S + T = S and S - T = S
So both of them are convex.
So answer is (E)
But i feel that the answer is too simple...and seems that i wrongly interpreted the question...
Hey! :o
We have the function $\displaystyle{y=f(x_1, x_2)=x_1\cdot x_2^2}$ and the set $M_1=\{x\in [0, \infty)^2 \mid f(x_1, x_2)>1\}$.
I want to check if the set is convex. Let $x=(x_1, x_2) , y=(y_1, y_2)\in M_1$, then $x_1\cdot x_2^2>1$ and $y_1\cdot y_2^2>1$.
We want to show that...
I need help on this problem:
If $X$ and $Y$ are convex sets, show that $X-Y = Z = \{x-y \mid x \in X, y \in Y\}$ is also convex.
Here are the steps I have gone so far:
Let $p \in Z$ such that $p = x_1 - y_1$, and let $q \in Z$ such that $q = x_2 - y_2$. Assume that $r$ lays in the segment...
Homework Statement
If the focal length of the bi convex lens is 25 cm then the power of the lens will be?
Since the light striking the lens converges at a point behind the lens will the answer come in positive?
Homework Equations
Power of lens D= 1/f
The Attempt at a Solution
D= 1/25= +0.04...