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. A

    Countable Non-Differentiable Points on Convex Curve Boundaries

    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...
  2. C

    Convex function on closed interval?

    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
  3. N

    Is a convex subset of a connected space connected?

    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...
  4. E

    Is the statement 'If E is an open connected set then it is convex' true?

    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...
  5. K

    How Does Strict Convexity Influence the Gradient Inequality?

    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...
  6. A

    Divide convex polygon into 4 equal areas

    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...
  7. K

    Convex set and convex functions

    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...
  8. A

    Focal length calculations for a double convex lens?

    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...
  9. Y

    What is the Focal Length of a Convex Lens?

    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...
  10. R

    Second derivative positive implikes midpoint convex

    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...
  11. P

    Is it possible to focus ultraviolet light with a convex spherical lens?

    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
  12. fsonnichsen

    Comparing GRIN Lenses vs Convex for Laser Beam Collimation

    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...
  13. T

    Shortest Distance between 2 convex sets

    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...
  14. J

    How to get the derivative of this convex quadratic

    \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?
  15. J

    How Does One Compute the Derivative of a Convex Quadratic Function?

    \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?
  16. J

    Prove intersection of convex cones is convex.

    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.
  17. R

    How to proove that that e^x is convex

    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...
  18. Z

    Is Every Convex Polytope Both Convex and Closed?

    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...
  19. Fredrik

    Probability measures and convex combination

    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...
  20. Y

    Difference between convex lens formula and convex mirror formula

    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...
  21. F

    Exploring the Relationship Between Convex Lenses and Objects

    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...
  22. F

    Does Focal Length Differ for Convex Lenses?

    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
  23. I

    Can spherical aberration be avoided in lenses with non-spherical curves?

    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...
  24. S

    Convex set for similarity constraint

    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...
  25. zimo

    A lens and a convex mirror problem

    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...
  26. O

    Prove Open Balls in an NLS are convex.

    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).
  27. W

    Convex set : characteristic cone

    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.
  28. I

    Diamond Ring in Front of a Convex Lens?

    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...
  29. D

    Finding Convex Lens Focal Length

    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...
  30. S

    Why Is an Image at Infinity Called Highly Magnified?

    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...
  31. G

    Radius of convex spherical mirror?

    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...
  32. A

    Convex mirror, reduced image, find radius of curvature

    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...
  33. B

    Increasing Laser Tag Gun Range with Convex Lenses?

    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...
  34. C

    Understanding Convex Analysis: Solving a Sequence of Sets

    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...
  35. J

    Affine Function on an Open Convex Set

    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."...
  36. M

    Proving f(z)=e^(g(z)) on a Convex Set Omega

    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...
  37. J

    Find focal length for meniscus convex

    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...
  38. M

    Convex Lenses: Determining Real/Virtual Images

    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 -...
  39. M

    Ray Diagram: Small Bulb & Convex Lens

    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...
  40. H

    Finding Convex Hulls in hyperbolic spaces

    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...
  41. C

    Finding the thickness of a double convex lens

    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...
  42. R

    Proving Finite Convex Sets Intersection is Convex

    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) <...
  43. B

    Convex loss function & normal posterior - Bayes's rule?

    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)...
  44. A

    Deriving the lens formula for a convex lens-say

    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...
  45. R

    Show that affine functions are both concave and convex

    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
  46. R

    Optimization proof for Ax > b. Prove that set is convex

    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...
  47. Z

    Pulleys + Convex Mirror Reflection

    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...
  48. C

    Optics problem: concave or convex

    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...
  49. D

    What is the proof that M is a convex and closed subset of C[0,1]?

    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}...
  50. V

    Hertz Contact Solution of Elastic Theory for Concave to Convex Shapes

    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...
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