# 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. ### 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. ### A Question regarding proof of convex body theorem

Hello, I am currently working on the proof of Minkowski's convex body theorem. The statement of the corollary here is the following: Now in the proof the following is done: My questions are as follows: First, why does the equality ##vol(S/2) = 2^{-m} vol(S)## hold here and second what...
3. ### 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. ### 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. ### Proof involving convex function and concave function

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

I need help on this problem: If $X$ and $Y$ are convex sets, show that $X-Y = Z = \{x-y \mid x \in X, y \in Y\}$ is also convex. Here are the steps I have gone so far: Let $p \in Z$ such that $p = x_1 - y_1$, and let $q \in Z$ such that $q = x_2 - y_2$. Assume that $r$ lays in the segment...
7. ### 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...
8. ### 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...
9. ### 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. ### 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. ### MHB Convex function and convex set

Convex function and convex set(#1 edited) Please answer #4, where I put my questions more specific. Thank you very much! The question is about convex function and convex set. Considering a constrained nonlinear programming (NLP) problem \[min \quad f({\bf x}) \quad {\bf x}\in \mathbb{R}^{n}...
12. ### Is this sufficiently proven? ( A set being a convex set)

Hi, just a few details prior: I'm trying to study techniques for maths proofs in general after having completed A level maths as I feel it will be of benefit later when actually doing more advanced maths/physics. With this question what is important is the proof is correct which means I don't...
13. ### MHB Inequality proof - for determining convex set

I am stuck at the inequality proof of this convext set problem. $\Omega = \{ \textbf{x} \in \mathbb{R}^2 | x_1^2 - x_2 \leq 6 \}$ The set should be a convex set, meaning for $\textbf{x}, \textbf{y} \in \mathbb{R}^2$ and $\theta \in [0,1]$, $\theta \textbf{x} + (1-\theta)\textbf{y}$ also belong...
14. ### Any compact subset is a contained in finite set + a convex set?

Homework Statement So I am trying to understand this proof and at one point they state that an arbitrary compact subset of a Banach space, or a completely metrizable space is the subset of a finite set and an arbitrary convex neighborhood of 0. I've been looking around and can't find anything...
15. ### Prove the following is a convex set?

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. ### Prove the Int<ABC is a convex set.

Homework Statement Prove the Int<ABC is a convex set. Homework Equations The Attempt at a Solution 1. Int <ABC = H(A,BC) intersect H(C,AB) by the definition of interior. 2. H(A,BC) is convex and H(C,AB) is convex by Half-Plane Axioms I know I need to show the intersection of...
17. ### Is the interior of an angle a convex set?

I need to prove the interior of <ABC is a convex set. I know it is. I started by defining the angle as the intersection of two half planes and using the fact that each half plane is convex. I am stuck on where to go from here.
18. ### 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. ### 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...
20. ### 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.
21. ### 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."...
22. ### 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...
23. ### 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.
24. ### Normed vector space: convex set

Homework Statement Show that the closed unit ball {x E V:||x||≤1} of a normed vector space, (V,||.||), is convex, meaning that if ||x||≤1 and ||y||≤1, then every point on the line segment between x and y has norm at most 1. (hint: describe the line segment algebraically in terms of x and y...
25. ### Proving a convex function on an open convex set satisfies some inequalities

Homework Statement Let f:\mathcal{O}\subset\mathbb{R}^n\rightarrow\mathbb{R}, \mathcal{O} is an open convex set. Assume that D^2f(x) is positive semi-definite \forall x\in\mathcal{O}. Such f are said to be convex functions.Homework Equations Prove that f((1-t)a+tb)\leq...
26. ### 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. ### 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...
28. ### 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...