What is Convex: Definition and 301 Discussions

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.

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  1. TheCapacitor

    Finding focal point of concave lens using concave and convex

    Homework Statement [/B] I was doing http://stao.ca/VLresources/sci-tie-data/lessons/1400_1499/DivergingLensExperimentDeta.pdf experiment. Let's look at this image: Suppose we get the imaginary object at dv by convergence lens. And this object is like a real object for the concave lens. Then...
  2. Y

    Why do convex lenses magnify, and why don't concave lenses?

    So knowing that convex lenses causes light to converge and concave lenses causes then to diverge, why does converging light create a larger image than diverging light? Magnification means we want to see more of a certain part of an object, meaning we want to enlarge that section and have it...
  3. B

    Calculating Image Height of Flower in Convex Mirror | Homework Question

    1. Homework Statement A convex spherical mirror with a focal length of magnitude 25 cm has a 4.0-cm tall flower placed 100 cm in front of it. What is the height of the image of the flower? a.20 cm b. 1.6 cm c .0.80 cm d. 4.0 cm e. 8.0 cm Homework Equations 1/f=1/do+1/di M=-di/do=hi/ho The...
  4. C

    Combing Lasers To Create One Mega Laser

    Hello, I am pretty new to physics but have a large amount of experience with Calculus and other sciences like chemistry. The theory I have come up with is that if I combine 6-8 5w laser beams together in a convex lens that can be moved forward or backward to change the focal point distance, I...
  5. F

    How do I show that a subset is closed and convex?

    We have a vector p = (0, 0, 2) in R^3 and we have the subset S = {xp where x >= 0} + T, where T is the convex hull of 5 vectors: (2,2,2), (4,2,2), (2,4,2), (4,4,6) and (2,2,10). How do I show that the subset T is a closed and convex subset? I know that a subset is called convex if it contains...
  6. A

    Forming an Image with a Convex Lens

    Using a convex lens, can an image be formed at the focal point and if so where would the object have to be from the convex lens?
  7. N

    A Certain convex minimization problem

    Hi, I would like to know if the inequality sign plays any role to the following optimization problem: minimize f0(x) subject to f1(x)>=0 where both f0(x) and f1(x) are convex. The standard form of these problems require a constraint such as: f1(x)<=0, but i am interested in the opposite...
  8. M

    Help with lenses for hand-made telescope

    I'm trying to make a handmade telescope, but I'm stuck with the lenses I bought 2 lenses. they are the same "convex from one side and concave from the other side" like glasses lenses. then I knew that I need a double convex for both the object lens and the eyepiece lens. so I decided to put the...
  9. M

    I Partitions of Euclidean space, cubic lattice, convex sets

    If the Euclidean plane is partitioned into convex sets each of area A in such a way that each contains exactly one vertex of a unit square lattice and this vertex is in its interior, is it true that A must be at least 1/2? If not what is the greatest lower bound for A? The analogous greatest...
  10. S

    Image formation by convex lens

    What will happen if look at(or in the direction of) a distant object through a convex lens?
  11. N

    Convex Lens: Image Formation at Infinity & Parallel Rays

    For an object at focal length in front of a convex lens, the image formed will be at infinity and the refracted rays are parallel (refer to the figure). Since our eye lens is convex in nature, why don't it converge these parallel rays so that an image can be formed?
  12. 0

    Determining the light intensity in a double convex lens

    Attached is an illustration of an image being demagnified using a double convex lens. I want to determine the intensity of the light (mW/cm2) if I were to put a lux meter at various locations after the convex lens given that the object is at a fixed location. The object is a monochromatic light...
  13. evinda

    MHB Is the Set K Convex and Does It Attain Its Euclidean Distance?

    Hello! (Wave) We consider the set $K=\{ x \in \mathbb{R}^n: \sum_{j=1}^n |x_j|^2 \leq 1 \}$ and let $ \in \mathbb{R}^n$. Check if the euclidean distance of $z$ from $K$ is attained. I want to show that $K$ is closed and convex. Then by a theorem, the distance will be attained by a unique...
  14. Siddharth Venu

    Convex Lens Problem: Find Focal Length

    A screen is placed 90cm from a object. The image of the object on the screen is formed by a convex lens at two different locations separated by 30cm. The focal length of the lens is what?
  15. noowutah

    Stationary point for convex difference measure

    Let \mathbb{S}^n be a simplex in \mathbb{R}^{n+1}, so \mathbb{S}^{n}=\{x\in\mathbb{R}^{n+1}|\sum{}x_{i}=1\}. Let D be a difference measure on \mathbb{S}^{n} with D(x,x)=0 and x=y for D(x,y)=0. D is also smooth, so differentiable as much as we need. Let (R) be a convexity requirement for D...
  16. heavystray

    Thick convex mirror or a thin one for the anti-theft mirror?

    the question in my exam goes like this: should we use a thick convex mirror or a thin one for the anti-theft mirror? the given answer: thin, because a thin convex mirror does not form multiple images can someone please explain to me why a thin spherical mirror does not form multiple images...
  17. heavystray

    Does Convex Mirror Thickness Affect Focal Length and Image Sharpness?

    hello guys, so i want to ask whether the thickness of a convex/concave mirror affect its focal length and the sharpness of image formed? and what really determines the sharpness of image? thanks for replying in advance! :biggrin::biggrin:
  18. N

    I Double slit particle distribution with convex lens?

    In one double slit experiment, one physicist placed a convex lens before the back screen detector. The individual photons passed through the double slit then through the lens. The individual photons then go through the focal point, then a good distance after they reach the focal point they hit...
  19. P

    Convex polyhedron - average of vertices inside polyhedron?

    I am interested in polyhedrons (mostly hexahedrons and pentahedrons). The shapes I am interested in are irregular, where none of the opposing faces are parallel to each other. However, the shapes I am dealing with all CONVEX. I have the vertices of my polyhedrons and was wondering if I sum up...
  20. M

    MHB What Are the Properties of Concave/Convex Spherical Mirrors?

    Spherical mirrors. We are given p= +18 and the mirror is concave with f= 12. Find r, i, m, Real or Virtual, Imaginary or not imaginary, and what side of the mirror is the image on. All units are cm. r: f=r/2, so r=2f r= 24 cm i: (1/p) + (1/i)= (1/f), so (1/f)-(1/p)=(1/i) i= 36 cm m...
  21. R

    MHB Finding Two Points in a Convex Set: Help Needed!

    I have two a convex set: {(x1, x2): 1≤ ∣x1∣ ≤2, ∣x2−3∣ ≤ 2} I have to find two points in the set for which the line segment joining the points goes outside the set. I have graphed the function and found my convex set. My question is, how do I find these two points? I have found various points...
  22. J

    Help please -- convex lens problem and focul length

    Homework Statement Suppose the convex lens is placed at the 49.4 cm mark, and an image forms on the screen at the 70.5 cm mark. Find: A) The focal length of this lens: ____ cm B) the absolute value of the magnification of this lens: ________ C) the height of the image: ________cm Homework...
  23. S

    How to Obtain a Magnification of -2 from a Convex Lens?

    Homework Statement You wish to obtain a magnification of -2 from a convex lens of focal length f. The only possible solution is to: A) place a virtual object at a distance 2f/3 from the lens. B) place a virtual object at a distance 5f/3 from the lens. C) place a real object at a distance...
  24. DrewHizzy

    Diverging and Converging lenses in a system

    Homework Statement A)Find the final position of the image (from the object, I assume?) B)Find the size of the final image of the object. Homework Equations 1/f = 1/do + 1/di The Attempt at a Solution 1) Solved first distance: 1/5 - 1/4 = 1/di --> 4/20 - 5/20, di = -20cm 2) Use di1 for do...
  25. X

    Convex mirror find object distance

    Homework Statement An object is placed in front of a convex mirror with a 45.6-cm radius of curvature. A virtual image half the size of the object is formed. At what distance is the object from the mirror? Homework Equations 1/f=1/d0+1/di hi/h0=-di/d0 The Attempt at a Solution hi=-1/2h0...
  26. C

    Intuition and existence for convex neighbourhoods

    Im trying to get some intuition for convex neighbourhoods which is neighbourhoods ##U## such that for any two points ##p## and ##q## in U there exists a unique geodesic connecting ##p## and ##q## staying within ##U##. QUESTION 1: These kind of neighbourhoods can be shown to always exist for...
  27. C

    Convex Neighbourhoods in Relativistic Spacetimes

    In general relativistic spacetimes, convex neighbourhoods are guaranteed to exist. As a reminder: a convex neighbourhood ##U## is a neighbourhood ##U## such that for any two points ##p## and ##q## in U there exists a unique geodesic connecting ##p## and ##q## staying within ##U##. With that...
  28. L

    MHB Face of a simplex, convex hull

    For a point $p$ in a simplex we define $ \text{supp (p)}= \{ i \in I_n \ | \ p_i \neq 0\}$ For a convex set $C$ we define $F$ to be its face if $F$ is convex and $\forall x,y \in C, \lambda \in (0,1) : \lambda x + (1- \lambda ) y \in F \Rightarrow x,y \in F$ For $A \subset \{1,2,...,n \}$ we...
  29. perplexabot

    Op-Amp convex optimization question

    Hey all. I have a question regarding an equation in a document I am reading about convex optimization of OP-AMPs. The document is attached, the equation is of section 3.4 and the equation number is (9). Can you please explain how it came to be? I understand that i3 = i4 = i5/2 but I don't...
  30. B

    Adjacent vertices in convex polygons

    While reading a bit about dihedral groups, I encountered a curiosity regarding convex polygons that I'm not sure is true or false. Given a convex polygon P, let A and B be adjacent vertices of this polygon and let C be a vertex of P not adjacent to A. Then is it necessarily the case that...
  31. S

    MHB Faces of a simplex, convex hull of the canonical basis of \$\mathbb{R}^n\$

    Let $I_n := \{1,2,...,n \}, \ p \in \Delta_n = \{(p_1, ..., p_n) \ | \ p_i \ge 0, \sum_{i=1}^n =1\}$ $ \text{supp (p)}= \{ i \in I_n \ | \ p_i \neq 0\}$ For a convex set $C$ we define $F$ to be its face if $F$ is convex and $\forall x,y \in C, \lambda \in (0,1) : \lambda x + (1- \lambda ) y...
  32. Prashasti

    Finding Focus Length in Broken Double Convex Lens

    Homework Statement A thin double convex lens of focal length f is broken down into two halves at the axis. The two haves are then combined, as shown in the figure. What will be the focal length for the arrangements...
  33. grandpa2390

    Objects in Mirror closer than appear Concave or Convex

    Homework Statement When a T.rex pursues a jeep in the movie Jurassic Park, we see a reflected image of the T.rex via a side-view mirror, on which is printed the (then darkly humorous) warning: “Objects in mirror are closer than they appear.” Is the mirror flat, convex, or concave? Homework...
  34. perplexabot

    Determining Convexity: S2 and Operations that Preserve Convexity

    Homework Statement Show if the set is convex or not! S2 = Homework Equations I know that to show a set is convex you can either use the definition or show that the set can be obtained from known convex sets under operations that preserve convexity. Convex definition: x1*Theta + (1 -...
  35. F

    Intersection of a closed convex set

    Let X be a real Banach Space, C be a closed convex subset of X. Define Lc = {f: f - a ∈ X* for some real number a and f(x) ≥ 0 for all x ∈ C} (X* is the dual space of X) Using a version of the Hahn - Banach Theorem to show that C = ∩ {x ∈ X: f(x) ≥ 0} with the index f ∈ Lc under the...
  36. A

    Proof that a TVS is seminormed iff it is locally convex

    Hi, There is a guided exercise in the course of functional analysis that I am following where we have to prove that a topological vectors space is seminormed if and only if it is locally convex. There is one step in te proof that I can't figure out. Let X be a topological vector space and...
  37. AwesomeTrains

    Ray transfer matrix - Plano convex lens

    Homework Statement I have to find the ray transfer matrix for a Plano convex lens, with a 5 cm radius and refraction index n=1.8. I'm supposed to use the paraxial approximation. Attached a homemade picture of the lens. Homework Equations Refraction at a curved surface: \begin{pmatrix} 1 & 0\\...
  38. AdityaDev

    Can a Convex Lens Converge a Tilted Wavefront?

    If I have a parallel beam of light parallel to x-axis, with the speed of light varying from bottom to top. Let it decrease from bottom to top. (To get such a beam one can pass the beam through a glass slab normal to its surface, whose refractive index increases linearly from bottom to top). Now...
  39. AdityaDev

    Modified plano-convex lens doubt

    I was thinking about the situation given my text about a plano convex lens which was produced with a manufacturing defect. It's plane surface is tilted outwards by a small angle 'z'. In the text its written that when a parallel light beam enters the lens parallel to x-axis , it will still be...
  40. T

    Ray Tracing for Convex Lens: How to Determine Image Distance and Realness

    Homework Statement [/B] B) Is the image real or virtual? Explain. C) Using the lens equation, find the image distance. 2. Homework Equations 1/di + 1/do = 1/f The Attempt at a Solution A) B) The image is real because it is on the opposite side of the lens as the source of light. C) do =...
  41. M

    Finding the Normal Cone of a Closed Convex Subset in a Hilbert Space

    Let \text{ } H \text{ }be \text{ }a \text{ } Hilbert \text{ } space, \text{ }K \text{ }be \text{ }a \text{ }closed\text{ }convex\text{ }subset \text{ } of \text{ }H \text{ }and \text{ }x_{0}\in K. \text{ }Then \\N_{K}(x_{0}) =\{y\in K:\langle y,x-x_{0}\rangle \leq 0,\forall x\in K\} .\text{...
  42. M

    Image seen by human eye through a convex lens

    Greetings, I don't have a magnifying glass/convex lens otherwise I would try to confirm this by experiment. Maybe I'll get one soon but to get things started. Let's say you have an object, with its own light source, and a convex lens with focal point f. The object is beyond 2f away from the...
  43. Z

    Coana-Effect over a convex surface-causes(liquid and gas flow)

    hello first of all i'd like to appologise for eventual speeling mistakes i am going to make in this thread :D ive chosen as subject for my maturity paper the coanda-effect applied in the common experiment with water flowing along the curved-convex surface of a spoon. i want to make a model of...
  44. DavideGenoa

    A separation theorem in locally convex spaces

    Dear friends, my book (an Italian language translation of Kolmogorov-Fomin's Элементы теории функций и функционального анализа) proves the following separation theorem: let ##A## and ##B## be convex sets of a normed space and let ##A## have a non-empty algebraic interior ##J(A)## such that...
  45. B

    Focal Length of a Plano Convex Lens

    Hi, The focal length of the Plano-Convex lens as calculated by the lens maker equation: Does it work when light crosses the lens from the convex side, planar side, or both? Thanks.
  46. berkeman

    Looking for an aftermarket convex backup mirror for vans/SUVs

    I saw a clever aftermarket rear backup mirror on a minivan recently, and I'd like to buy one for my Suburban. It's about 9" in diameter, and attaches to the rear door of a minivan or SUV at the door seam along the side near the top. This makes it visible through the rear window, so you can...
  47. B

    Convex mirror question how long the image is?

    Shiny lawn spheres placed on pedestals are convex mirrors. One such sphere has a diameter of 40 cm. A 12 cm long robin sits in a tree 1.5 m from the sphere. (a) Where is the image of the robin? (b) How long is the image? f=R/2 to find radius 1/v=1/f-1/u to find image distance (the mirror...
  48. D

    MHB Convex and concave lens and paraxial rays

    What effect would a convex (concave) lens have on paraxial rays? For a convex lens, the rays would pass through and converge on the focus on the other side. For a concave lens, the rays would reflect back to the focus on the same side. Is this correct?
  49. J

    Solving a convex mirror problem

    Homework Statement Given the following information: convex mirror, focal length (f) = 12.5 metres, height object (ho) = 5 metres, height image (hi) = 1 metre. Find distance object (do) and distance image (di). Homework Equations 1/f = 1/do + 1/di Magnification= -hi/ho = di/do...
  50. Racer_Rob

    Focusing laser beams with concave and convex lenses

    I'm trying to focus a 3mm diameter laser beam (527nm) to a 0.3mm spot size. According to this online calculator I need a +500mm focal length lens and a -50mm FL lens separated by 450mm. However, in practice this did not produce a focused spot. I'm assuming this calculator has made some...
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