What is Convex set: Definition and 28 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. P

    I On a proof of the converse of the supporting hyperplane theorem

    The converse of the supporting hyperplane theorem states Here's the "proof": I've been told that any proof that does not use the fact that ##C## has non-empty interior will not work, because it easy to construct counterexamples of sets that will fail if they have empty interior. I'm not sure...
  2. P

    A Question regarding proof of convex body theorem

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  3. B

    I Proving Convexity of the Set X = {(x, y) E R^2; ax + by <= c} in R^2

    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...
  4. Onezimo Cardoso

    How to Prove Inequality for Convex Sets in R^n?

    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...
  5. TyroneTheDino

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    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...
  6. R

    MHB Proving Convex Set Properties: A Guide to Showing the Convexity of X-Y

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  7. F

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

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  8. R

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

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  9. 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 -...
  10. 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...
  11. B

    MHB Convex function and convex set

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  12. D

    Is this sufficiently proven? ( A set being a convex set)

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  13. N

    MHB Inequality proof - for determining convex set

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  14. F

    Any compact subset is a contained in finite set + a convex set?

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  15. A

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    Homework Statement Prove that F = {x E R^n : Ax >/= b; x >/= 0} is a convex set. Yes x in non negative and A and b are any arbitrary Homework Equations The Attempt at a Solution Well I know A set T is convex if x1, x2 E T implies that px1+(1-p)x2 E T for all 0 <= p <= 1...
  16. L

    Prove the Int<ABC is a convex set.

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  17. L

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  18. 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...
  19. S

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  20. W

    Convex set : characteristic cone

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  21. J

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    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."...
  22. M

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

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  23. Telemachus

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    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.
  24. K

    Normed vector space: convex set

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  25. M

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  26. 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...
  27. E

    Seperation of a Point and Convex Set

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  28. 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...
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