What is Convex: Definition and 302 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. X

    Determining Focal Length of Convex Lens Without Apparatus

    So I got a lab to plan and design an experiment to determine the focal length of a convex lens and involve the use of lens eq. and by means of plotting a graph...I have a basic rundown of what needs to be done and all but I am somewhat in a dilemma i don't have any access to any of the apparatus...
  2. T

    How Do You Calculate Image Position and Magnification with a Concave Mirror?

    I'm really bad at physics and am quite lost on a homework assignment of mine. Any help that I can get would be fantastic, thanks. Homework Statement A concave mirror has a radius of curvature of 60 cm. Calculate the image position and magnification of an object placed in front of the...
  3. B

    Variational methods - prove f is convex in R->R

    Suppose f:R^N -> R is twice differentiable. Prove that f is convex if and only if its Hessian gradiant^2 f(x) is nonnegative. How do I go about proving this? and my professor said I only need to consider when N=1. so R->R. any help would be greatly appreciated. For proving it backwards...
  4. B

    Variational methods - properties of convex hull

    Show the following properties of convex hull: (a) Co(CoA) = Co(A) (b) Co(AUB) \supseteqCo(A) U Co(B) (c) If A\subseteqB then Co(AUB)=Co(B) (d) If A\subseteqB then Co(A)\subseteqCo(B) The definition of a convex hull is a set of points A is the minimum convex set containing A. (c) is quite...
  5. S

    What is the Homeomorphism Proof for Non-Empty Convex Open Subsets of R^2?

    Homework Statement Let U be a non-empty, convex, open subset of R^2. Prove that U is homeomorphic to R^2. Hint: First prove that the intersection of a line in R^2 with U (if non-empty) is homeomorphic to an open interval in R^1. Then use radial projections.Homework Equations We just have the...
  6. E

    Seperation of a Point and Convex Set

    [SOLVED] Seperation of a Point and Convex Set Homework Statement Let C be a closed convex set and let r be a point not in C. It is a fact that there is a point p in C with |r - p| l<= |r - q| for all q in C. Let L be the perpendicular bisector of the line segment from r to p. Show that no...
  7. D

    Average of Log of a Function: Bounded by 1 and Convex

    Hello, I am interested in the average behaviour of the log of a function. I know the average of the function over the range of interest: F = \frac{1}{(b-a)} \int_a^b f(x) dx. I also know that f(x) is convex and bounded from below by 1. I want to know the average \frac{1}{(b-a)}...
  8. quasar987

    Little bit of convex analysis on a Hilbert space

    [SOLVED] Little bit of convex analysis on a Hilbert space Homework Statement Let H be a Hilbert space over R and f:H-->R a function that is bounded below, convex and lower semi continuous (i.e., f(x) \leq \liminf_{y\rightarrow x}f(y) for all x in H). (a) For all x in H and lambda>0, show that...
  9. Z

    Solving Refraction Puzzles: Coin Depth & Convex Lens Image

    Problem 1: A thin hollow glass sphere has a radius of 1 m. A coin is placed at the bottom of the glass sphere. 2.09 metre cube of a liquid of refractive index 1.5 is poured into the glass shpere. A person is viewing the object vertically. What is the apparent depth of the coin? Im not sure...
  10. D

    Graphically interpreting data regarding magnification with a convex lens

    Homework Statement How do I find the relationship of magnification to the graph of "Object Distance vs. Image Distance" or the graph of "Inverse Object Distance vs. Inverse Image Distance" for data points collected from a light box and a convex lens? Basically for the lab I used a light...
  11. X

    Maximal area of a convex region bounded by hyperbolas

    This problem was suggested by Gokul43201, based on this year's Putnam A2. Suppose that K is a convex set in \mathbb{R}^2 which is contained in the region bounded by the graphs of the hyperbolas xy=1, xy=-1 (so the set is in the inner + shaped region which contains the origin also). What is...
  12. L

    Proving a Convex Quadrilateral is a Square with Internal Point O as its Center

    Homework Statement Let O be an internal point of a convex quaddrilateral PQRS whose area is A. Prove that, if 2A = OP^2 + OQ^2 + OR^2 + OS^2, then PQRS is a square with O as its centre Homework Equations The Attempt at a Solution I have no idea where to start, except that I...
  13. quasar987

    Real analysis - show convex functions are left &amp; right differentiable

    [SOLVED] Real analysis - show convex functions are left &amp; right differentiable Homework Statement Let f:R-->R be convex. Show f admits in every point a left derivative and a right derivative. Homework Equations A function f:R-->R is convex if x1 < x < x2 implies f(x)\leq...
  14. L

    An algebraic proof that a convex quadrilateral has intersecting diagonals

    Looking for an algebraic proof that a convex quadrilateral has intersecting diagonals So I'm trying to find an algebraic proof that the diagonals of a convex quadrilateral intersect because I'm working on a proof of a generalization of this idea into higher dimensions and I really have no idea...
  15. G

    How Does Magnification Change When Switching from Convex to Concave Mirror?

    A spherical mirror is polished on both sides. When used as a convex mirror, the magnification is +1/4. What is the magnification when used as a concave mirror, the object remaining the same distance from the mirror? I started with: mconvex = 1/4 mconcave = ? doconvex = doconcave...
  16. S

    About convex hull and fixed point

    First, I wonder whether I can put the post here... Given X=[0,1]^2 a(x)={y in X:||y-x||>=1/4} b(x)is the convex hull of a(x). Identify the set of fixed points. My answer is 3/4>=x>=1/4, 3/4>=y>=1/4, but I am not sure... What if we have a(x)={y in X:||y-x||>=1/2}? (My answer is...
  17. S

    About convex hull and fixed point

    X=[0,1]^2 a(x)={y in X:||y-x||>=1/4} b(x)is the convex hull of a(x). Identify the set of fixed points. My answer is 3/4>=x>=1/4, 3/4>=y>=1/4, but I am not sure... Thanks.
  18. L

    Lattice points : Convex region symm. about the origin

    Let R be a convex region symmetrical about the origin with area greater than 4. Show that R must contain a lattice point different from the origin. This is the 2-D case of Minkowski's theorem, right ? How about the n-dimensional version ? The n-dimensional version is : Given a convex...
  19. Alpha Russ Omega

    Reflecting glass and a convex mirror.

    Hello! I'm stuck on this problem... A thin flat plate of partially reflecting glass is a distance (b) from a convex mirror. A point source of light (S) is placed a distance (a) in front of the plate so that its image in the partially reflecting plate coincides with its image in the mirror...
  20. M

    A point of a closed convex set?

    Homework Statement Given D a a closed convex in R4 which consists of points (1,x_2,x_3,x_4) which satisfies that that 0\leq x_2,0 \leq x_3 and that x_2^2 - x_3 \leq 0 The Attempt at a Solution Then to show that either the point a: = (1,-1,0,1) or b:=(1,0,0,-1) is part of the...
  21. M

    Maple Drawing a Convex Hull in Maple

    Urgent Question: Drawing a Convex Hull in Maple Homework Statement Drawing a Convex Hull of 5 2D Points... Homework Equations None The Attempt at a Solution Hi there, I am trying draw the convex hull of the 5 points x1,x2,x3,x4,x5 below. Is this the correct way of doing...
  22. M

    Sun's Rays & Convex Mirrors: What is the Image Size?

    Rays from the sun subtend an angle theta (in radians) at the pole of a convex mirror (in radians) at the pole of a mirror of focal length f. If the diameter of the sun is D ,then diameter of the image of the sun formed by the mirror is: (a) f theta (b) 2 f theta (c) 6 f theta (d) none...
  23. A

    Solving for x in a Combination of Two Convex Lenses

    Homework Statement A combination of two thin convex lenses of equal focal lengths,is kept separated along the optic axes by a distance of 20 cm between them.The combination behaves as a lens system of infinite focal length.If an object is kept at 10 cm from the first lens,its image...
  24. T

    Solve Convex Lens Question: Find Focal Length & Magnification

    An object and a screen are fixed at a distance of 80cm apart and a convex lens forms a real image of the object on the screen. When the lens is moved along its axis a distance of 16cm, a real image of the object is again formed on the screen. Find the focal length of the lens and the...
  25. B

    Discover the Focal Length of a Convex Mirror Using Object and Image Distance

    Convex mirror. PLEASE HELP! Homework Statement An object 30 cm tall is placed 20 cm in front of a convex mirror. If the height of the image is +10 cm, find the focal length of the mirror. Homework Equations The Attempt at a Solution I know to use 1/f = -(Image distance from V)...
  26. A

    Convex Refraction: Why do objects appear larger when viewed through a fishbowl?

    Homework Statement Why do fish appear bigger when you look at them through a fishbowl? Homework Equations The Attempt at a Solution because objects in water appear bigger than they actually are? this has to do with change in the medium from air n1 to water n2?
  27. B

    Solving Final Image Formed from Convex and Concave Lens

    Homework Statement An object is placed 6 metres from a convex lens of focal length 30cm. A concave lens of focal length 5cm is then placed 20 cm from the concave lens, on the side distant from the object. Determine the position, magnification and nature of the final image formed (to solve...
  28. B

    Is f convex on (a,b) if and only if f(y)-f(x)>(y-x)f'(x) for all x, y in (a,b)?

    Can someone tell me if my logic in answering the following questions is ok: 1. Suppose that f(x) and g(x) are convex on (a,b). Show that the functions h(x)=max[f(x), g(x)] is also convex on (a,b). -I said that since f and g are convex their second derivatives are not equal to zero in (a,b)...
  29. N

    Radius of Curvature of Convex Mirror: -17.39 cm

    A real object is placed at the zero end of a meterstick. A large concave mirror at the 100 cm end of the meterstuck forms an image of the object at the 82.4 cm position. A small convex mirror placed at the 20 cm pisition form a final image at the 6.3 cm point. What is the radius of curvature of...
  30. MathematicalPhysicist

    A question regarding convex function.

    let f:(a,b)-> a differentiable function, f is a convex function iff for every x in (a,b) the tangent line to f's graph in x is below f. i tried it this way: suppose f is a convex function, then for every 0<d<1 and every s,t in (a,b), f(dt+(1-d)s)<=df(t)+(1-d)f(s), now the tangent line in...
  31. Amith2006

    Focal length of plano convex lens

    # Diameter of a Plano-convex lens is 6 cm and the thickness at the center is 3 mm. If the speed of light in the material of the lens is 2 x 10^8 m/sec, what is the focal length of the lens? I solved it in the following way: Let R be the radius of curvature of the convex surface. Given that AB...
  32. Amith2006

    Focal length of concave lens using convex lens

    # A convex lens of focal length 10 cm forms a real image of an object placed at a distance of 20 cm from it. Midway between the convex lens and the position of the image a thin concave lens is introduced. The image formed now is at a distance of 5 cm away from the earlier position. What is the...
  33. A

    Convex Lens Image: Solve Problem & Describe Image

    I have to complete a problem about locating an image for a convex lens but I'm having trouble with it. I tried it out, but I can't complete it. Can someone help please? Locate the image for the convex lens below. Use any two rays to do so. Give the characteristics of the image. I'm having...
  34. Amith2006

    Image formed by a convex mirror

    Sir, I am posting these questions for the 3rd time as you didn't respond. Please respond. 1)A convergent beam of light is incident on a convex mirror of radius of curvature 60 cm as shown in figure. What is the nature and position of the image formed by it? I solved it in the following way...
  35. Amith2006

    Image formed by a convex mirror

    Sir, 1) A man 2 meters tall stands 5 meters in front of a large vertical plane mirror. What is the angle subtended by his image in the mirror at his eye? I solved it in the following way: Let AB represent the man and CE represent the mirror mounted on the wall.Here I am assuming that the...
  36. M

    Having trouble proving a property of convex sets:

    In my econ homework, I was asked to prove that: A set C is convex iff a C + b C = (a+b) C for all nonnegative scalars a and b. All that I'm given is that the definition of a convex set is, for x,y elements of a convex set C: (1-a) x + a y exists in C, for 0<a<1 My thoughts were to first...
  37. M

    Troubleshooting Convex Mirror: Image Distance & Magnification

    Hello everyone, I seem to be getting all the mirrors right that where concave but now i have a convex one and it doesn't seem to be working when im' finding the image distance and magnification/properties. Here is the problem: Heres my work: both the i and the m where wrong, i also tried...
  38. mattmns

    Lin. Alg. - Half Planes (convex)

    Hello, I am a little confused on how to prove these half planes are convex, and my book does not actually show an example. ---------- So here is a problem, my book briefly talks about it: Show that the plane 2x - 3y >= 6 is convex. So if we let A = (2,-3), and X = (x,y), then we can write...
  39. K

    Nonempty Convex Sets: Proving Closure

    Hi, Let C_1 and C_2 be nonempty convex sets and suppose C_1 \cap C_2 \neq \emptyset . I read a text that claims \textup{cl}(C_1 \cap C_2) \subset \textup{cl}(C_1) \cap \textup{cl}(C_2) since C_1 \cap C_2 \subset \textup{cl}(C_1) \cap \textup{cl}(C_2). I am able to prove the latter, but I...
  40. K

    If A and B are nonempty convex sets, and C = A + B, then..

    If A and B are nonempty convex sets. And C = A + B. How to prove int(C) = int(A) + int(B)?
  41. D

    Object magnification after Convex surface

    I have an object that is surrounded by air and sits a distance U=30 from a convex surface. The convex surface has a radius of R=5. The index of refraction of the refracting media is n=1.33. How do I determine the magnification of the object?
  42. P

    A question about the minimum/maximum of a convex function

    I would like to be sure in the following, not prove it, just have it confirmed... If a function f is convex, then it has 1.) only one maximum and no minimum 2.) only one minimum and no maximum infinity and -infinity are not included.
  43. S

    Is the Union of Two Intervals Convex?

    Hello! I've got some questions concerning convex sets. We've had a lecture about convex sets this week, and got a some basic problems to solve. I think I can't use the material in the lecture to solve the problem. I'm just not sure about whether I fully understand the concept and can use it...
  44. S

    At Center of Convex Lens & Glass Plate: What is Observed?

    A convex lens is placed on a flat glass plate and illuminated from above with monochromatic red light. When viewed from above, concentric bans of red and dark are observed. What does one observe at the exact center of the lens where the lens and the glass plate are in direct contact.
  45. S

    Hwk Problem about Convex Mirrors

    An object that is 31 cm in front of a convex mirror has an image located 20 cm behind the mirror. How far behind the mirror is the image located when the object is 19 cm in front of the mirror? For this problem, I know I have to use the mirror equation but I keep getting the wrong...
  46. S

    Physics Hwk Problem on Convex Mirrors

    Convex mirrors are being used to monitor the aisles in a store. The mirrors have a radius of curvature of 4 m. What is the image distance if a customer is 20 m in front of the mirror? From using the mirror equation, i came up with an answer of 1.82m but when i submitted it for webassign, it...
  47. N

    Help Solve Physics Problem Involving a Convex Lens

    Lez prob hi, An object is moving with velocity 0.01 m/s towards a convex lens of focal length 0.3m. Find the magnitude of rate of separation of image from the lens when the object is at a distance of 0.4 m from the lens. Also calculate the magnitude of the rate of change of the lateral...
  48. S

    Convex Mirrors magnification help

    I know that magnification of a single convex mirror is always positive (that is, greater than 0) based on f<0, but does the size of the magnification depend on the magnitude of the object distance?
  49. A

    Convex mirror, and you point a light at it horizontally

    If you have a convex mirror, and you point a light at it horizontally, does the light always reflect through the focal point? (the point halfway on the line connecting border and the center, parallel to the light's initial path) I remember learning this, but I think of a mirror in the shape...
  50. C

    Why Convex Mirrors are Used in Late Model Cars

    The rear view on the passenger side of late model cars warn us theat objects may be closer than they appear. I think the mirror used is convex. But why is that type of mirror selected? thanks joe
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