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. 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...
  2. M

    Difference between a convex norm and strong convex norme ?

    hi :) if someone have any idea ? what is the difference between a convex norm and strong convex norme ?
  3. S

    Convex lens questionCant find the equation i need. Please help

    A convex lens of focal point 0.8m focuses a real image of the moon onto a screen. If the moon subtends an angle of 0.5 degrees to an observer on earth, calculate the size of the moons image on the screen? Thats the question guys, I am really struggling to find the appropriate equation as we...
  4. A

    How Does Bisecting a Lens Affect Its Focal Length?

    1)Why when a convex lens having radius of curvature R, focal length f and refractive index μ is bisected horizontally along the principal axis its focal length remains the same, whereas when it is bisected vertically focal length becomes 2f? Does the same thing happens in the case of...
  5. D

    An inequality on the diagonals of a convex quadrilateral

    Hi, It is well known by Varignon's Theorem and the Triangle Inequality that in a convex quadrilateral ABCD with midpoints a of side AB, b of side BC, ... d of side DA and perimeter p, \[ AC + BD < p\] where AC and BD are the diagonals of the quadrilateral. However, how do I obtain...
  6. D

    An inequality on the diagonals of a convex quadrilateral

    Hi, It is well known by Varignon's Theorem and the Triangle Inequality that in a convex quadrilateral ABCD with midpoints a of side AB, b of side BC, ... d of side DA and perimeter p, AC + BD < p where $AC$ and $BD$ are the diagonals of the quadrilateral. However, how do I obtain...
  7. C

    Fitting a convex polytope into another convex polytope

    Hi, I am presently writing Python code to solve the following problem; fit the largest volume polytope (P1), with a given number of faces, into a known other polytope (P2) (with more faces than the one being fitted inside). Both polytopes are convex and are described by the feasible region of...
  8. C

    Exploring the Difference in Focal Lengths of Telephoto & Convex Lenses

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  9. B

    The position of the object is on the place of the question mark.

    Homework Statement A convex mirror of focal length 10cm forms an image that is one quarter the size of the object. Find the position of the object and the image. Homework Equations -1/f = 1/u + 1/v m = v/u The Attempt at a Solution m = 1/4 f = 10cm -1/10 = 1/u + 1/0.25u I think...
  10. Telemachus

    Topology: is this a convex set?

    Homework Statement Hi there, I have a set similar to this \{(x,y)\in{\mathbb{R}^2}:x^2+y^2\neq{k^2},k\in{\mathbb{Z}\} (its the same kind, but with elipses). And I don't know if it is convex or not. If I make the "line proof", then I should say no. What you say? Bye there, and thanks.
  11. C

    How Does Light Behave When Emitted From a Focal Point Through a Convex Lens?

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  12. H

    Monotonicity of convex function

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  13. K

    Normed vector space: convex set

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  14. K

    What is the formula for determining image distance in a convex mirror?

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  15. M

    How to Calculate Focal Length of a Convex Mirror?

    Homework Statement The virtual image produced by a convex mirror is one-quarter the size of the object. b) What is the focal length of this mirror? Homework Equations 1/di = 1/f - 1/do The Attempt at a Solution i found di to be -9.3 cm but when i solved for f i keep getting it...
  16. O

    How Do You Calculate Object-Image Separation for a Convex Mirror?

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  17. M

    Proving a convex function on an open convex set satisfies some inequalities

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  18. A

    Is Normal Reaction Higher on Concave Bridges Than Convex Bridges?

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  19. L

    Negative Focal Length and Convex Mirrors: Is 1/di Negative?

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  20. B

    Calculating image locations with thin lenses

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  21. N

    Convex Mirror Image Distance and Magnification

    Homework Statement At an intersection of hospital hallways, a convex mirror is mounted high on a wall to help people avoid collisions. The mirror has a radius of curvature of 0.510 m. -What is the image distance for a patient 11.9 m from the mirror? (Use the correct sign conventions.)...
  22. H

    Separation and Support Theorems for Convex Sets

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  23. M

    Solve Convex Function: Show yf(y^{-1}\textbf{x}) is Convex | Mathmos6

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  24. J

    About the number of triangles of convex n-gon

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  25. D

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  26. C

    Is the union of convex sets always convex?

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  27. K

    Convex Mirrors and Magnification

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  28. W

    Calculating Image Position and Size Using Convex Mirrors

    1. Shiny lawn spheres placed on pedestals are convex mirrors. One such sphere has a diameter of 40 cm. A 12 cm robin sits in a tree 1.5 m from the sphere. a) Where is the image of the robin? b) How long is the robin's image? 2. 1/f = 1/do + 1/di 3. a. 1/f = 1/do + 1/di...
  29. R

    Convex Mirror Image Calculation

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  30. I

    Why Is the Virtual Image in a Convex Mirror Always Smaller?

    Homework Statement Prove that the virtual image in a convex mirror is always smaller than the real object. Homework Equations m = -\frac{d_{i}}{d_{O}} The Attempt at a Solution Not a homework problem. Something which is bothering me, and haven't been able to prove yet. Thanks!
  31. L

    How Far is the Second Object from the Convex Mirror?

    Homework Statement An object is located 14 cm in front of a convex mirror, the image being 7 cm behind the mirror. A second object, twice as tall as the first one, is placed in front of the mirror, but at a different location. The image of this second object has the same height as the other...
  32. R

    [complex functions] polynomial roots in a convex hull

    Hi everyone, I've got this problem to solve: My problem is that I don't fully understand the question. I have found such definition of convex hull: So I do have to prove, that all the roots of W'(z) [let's denote them as z'_k] must be able to be written in such form: z'_k = \sum_{k=1}^n...
  33. N

    Solving for p: Finding Distance from Convex Lens

    Homework Statement Given only the distance between the object and its image (p +q), how is it possible to find p, the distance of the object from the convex lens? Homework Equations 1/p + 1/q = 1/f in this case, p + q =40 also, the image is smaller than the original, so p > 2f The...
  34. E

    Convex Subsets of Topological Vector Spaces

    I had a quick question: Is the following proof of the theorem below correct? Theorem: If C is a convex subset of a Topological vector space X, and the origin 0 in X is contained in C, then the set tC is a subset of C for each 0<=t<=1. Proof: Since C is convex, then t*x + (1-t)*y...
  35. E

    Convex mirrors - Sign convention trouble

    Homework Statement I solved the problem, but upon further observation, I discovered that what I did didn't make sense. (these are rounded numbers, which shouldn't make a difference in my question) do = 13 cm (do is what I solved for in the problem, and according to the online system it was...
  36. C

    Convex combination and lp's

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  37. N

    Inner-radius of a (convex) polygon

    Hi all, I want to know whether it is correct that every convex polygon has an inner-circle (& hence an inner-radius). I think it is only possible for triangle and for regular polygon. Am I right? If there is any convex N-gon having sides a_1,a_2,...,a_N which has an incircle, then...
  38. P

    Proving x* as an Extreme Point of a Convex Set | Homework Question

    Homework Statement Let x* be an element of a convex set S. Show that x* is an extreme point of S if and only if the set S\{x*} is a convex set. Homework Equations (1-λ)x1 + λx2 exists in the convex set The Attempt at a Solution I'm not too sure what S\{x*}, I asssumed it was...
  39. S

    Number of edges of a convex polytope with n vertices

    Homework Statement What is the number of edges of a convex polytope with n vertices all of whose faces are triangles. Homework Equations # of faces + #of vertecies = # of edges + 2 The Attempt at a Solution My reasoning is as follows: n/3 + n = # of edges + 2 4n/3 - 2 =...
  40. L

    Def'n of holomorphically convex domain

    Does anyone have a textbook on multivariable complex analysis, and do they define a holomorphically convex domain in terms of the union of an ascending series of compact subsets? If so, how exactly does the definition go? Was it D=\bigcup K_n where K_n is holomorphically convex and compact...
  41. S

    Prove S is Convex: Help with Convexity

    How do i prove this? Let S = {(X1, …., Xn) € R^n | Xi ≥0, X1 + … +Xn = 1}. Show that S is convex. Suppose f(Si) = {X1, X2,..., Xn} AND g(Si) = {X1 + X2+ ...+Xn} but If do that, then f(Si) and g(Si), both will increase. Now I'm not sure where to go from here. *PS: Both functions are...
  42. A

    Show Proving Convexity of f(x)=||x|| Function

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  43. S

    Proof of continuity of convex functions

    Homework Statement Let a function f : R => R be convex. Show that f is necessarily continuous. Hence, there can be no convex functions that are not also continuous. Homework Equations The Attempt at a Solution F is continuos if there exist \epsilon >0 and \delta>0 such that |x-y|<...
  44. B

    Convexity of continuous real function, midpoint convex

    Homework Statement Assume f is a continuous real function defined in (a,b) such that f(\frac{x+y}{2})<=\frac{f(x)+f(y)}{2} for all x,y in(a,b) then f is convex. Homework Equations The Attempt at a Solution my attempt is to suppose there are 3 points p<r<q such that f(r)>g(r)...
  45. G

    Magnification in a Convex Lens: + or -?

    In a conversing lens, if the object is not within the focal point, then the magnification is "-". But in the case where the object is placed in the focal point, then the magnification is "+"?
  46. G

    Creating an Image 2x Size of Object w/ Convex Lens: Half Focal Length Away

    Homework Statement A converging lens (convex lens) creates an image that is 2 times the size of the object. The object is placed: Homework Equations -di/do = hi/ho 1/f = 1/di + 1/do The Attempt at a Solution I thought the answer was that the object is place two focal lengths...
  47. N

    Convex Polyhedron: half-planes to triangular mesh

    Hello! I'm trying to find some information about my problem but it doesn't seem very easy. 1 - I have a convex polyhedron defined as the intersection of several half-planes. 2- Now I would like to obtain a triangularization of the polyhedron surface in the best way. Can anyone indicate me...
  48. J

    Find the focal length of a concave lens using a convex lens?

    Can anyone explain to me the procedure that one would follow to you find the focal length of a concave lens using a convex lens? Thanks.
  49. P

    Convex Lens and Light Refraction

    Hi all, if light was to go straight through a convex lens would it come out straight or inverted to a point? also if light was to go through the same lens but goes through going inwards how would the light be refracted?
  50. M

    Is f(x,y) = (x^2) * exp(y) a convex or concave function?

    Could anyone comment on the convexity of f(x,y) = (x^2) * exp(y) ... i.e. x square into e to the power y. I did try to find Hessian of the same and the value I get is : Hessian(x,y) = -2 * x^2 * exp(2y)... which looks <= 0 for all x and y. I assume this should imply f(x,y) is...
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