Convex set Definition and 27 Threads

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...

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...

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...

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...

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...

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...

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...

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...

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 -...

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...

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}...

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...

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...

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...

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...

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...

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.

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...

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...

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.

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...

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.

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...

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...

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...

[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...

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...