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Questions tagged [convex-analysis]

Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

Convex analysis is the study of properties of convex sets and convex functions.

Formally, a set $S$ is convex if, for all $x, y \in S$ and $t \in [0,1]$, $tx + (1-t)y \in S$. Intuitively, this says that for any two points in $S$, the line segment connecting those points is also in $S$.

Formally, a function $f: S \to \mathbb{R}$ defined on a convex set $S$ is convex, if for all $x,y \in S$ and $t \in [0,1]$, $f(tx + (1-t)y) \leq t f(x) + (1-t) f(y)$. Intuitively, this says that, for any two points on the graph of $f$, the line segment connecting those points lies above the graph of $f$.

Some of the more important results in convex analysis include Carathéodory's Theorem, Jensen's Inequality, Minkowski's Theorem, and the supporting hyperplane theorem.

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Is there some generalization of convex sets along these lines?

Consider subsets of $\mathbb{R}^n$. Instead of a straight line, every pair of points have to be joined by a line with the curvature of a parabola or less than it . This would include the straight line too. I guess one could arrive at some…
user2277550
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Prove that the linear combination of points in a convex set must be in the set

Let K be a convex set. Suppose $x_1,x_2,...,x_n \in$ K. Prove that: $$ x=\sum_1^n a_jx_j \in K, \space\space\space\space\space\space\space where \space\space\space\space\space\space\space\sum_1^na_j=1 $$ I tried to prove this by induction but failed…
A Slow Learner
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If there always a supporting hyperplane $H$ through an extreme point $x$ of a convex set $X$ such that $H\cap X=\{x\}$?

Let $X \subset \mathbb{R}^n$ be a nonempty convex set, and suppose $x$ is an extreme point of $X$. Does there always exist a supporting hyperplane $H$ to $X$ at $x$ (i.e., $x$ lies on $H$) such that $H \cap X = \{x\}$? Thoughts: My intuition says…
madnessweasley
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Prove whether $\{ x \in \mathbb R^n \mid a^T_1x \geq b_1 \lor a^T_2x \geq b_2\}$ is convex or not

Prove whether $$\{ x \in \mathbb R^n \mid a^T_1x \geq b_1 \lor a^T_2x \geq b_2\}$$ is convex or not. I think if $a_1=a_2$, $b_1=b_2$, then it is convex. It's not convex under other situations. Is that correct? Any help to prove that? Thanks a lot!
Matata
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equivalent definition of convex functions

I got the following definition of a convex functions on $\mathbb R$: Let $I \subseteq \mathbb R$ be an interval and $f \to \mathbb R$. Then $f$ is called convex, if for all $x_1,x_2 \in I$ and all $\lambda \in [0,1]$ holds that $f((1-\lambda)x_1 +…
Pazu
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Unions of convex functions

Let $A$ and $B$ denote open convex subsets of $\mathbb{R}^n$. Suppose $f$ and $g$ are convex functions on $A$ and $B$ respectively, mapping into the real line. It seems likely that if $f$ and $g$ agree on $A \cap B$, and if $A \cup B$ is convex,…
goblin GONE
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Any convex set is pre-image of a convex set under a convex function

Let $C\subseteq R^n$ be a compact convex set. Is there a convex function $f : R^n\to R$ and real numbers $a\leq b$, such that $C=f^{-1}([a, b])$?
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extreme point of a convex set is also a bounday point?

Could anyone help me to show: if C is a convex set and x is an extreme point of C then it is also a boundary point of C. Thank you.
Myshkin
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From concavity to convexity

Suppose I have a function $g$ on $[-1,1]$ which is increasing, concave and such that $g(-1)=0$. I set $h(t)=\frac{g(t)}{1+t}$. Is it true that $h$ is decreasing and convex?
Babyblog
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How to prove $\|\bf x\|^{\it p}_{\it p}$ can be a convex function on $ \mathbb R^n $

How to show that $ \|\mathbf{x}\|^p_p $ can be a convex function on $ \mathbb R^n $ for $ p>1 $, where $ \displaystyle \|\mathbf{x}\|^p_p = \sum_{i=0}^n |\mathcal{x_i}|^p $ is the pth power of the $ l_{p} $-norm of the vector $ \mathbf x \in…
Julie
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Projection onto closed convex set

Show that the function defined by $f(t)=|P_{D}(x+td)-x|$ is nondecreasing, where $D$ is closed convex, $x\in D$, $t\geq 0$, $d\in \mathbb{R}^{n}$ and $P_{D}$ is projection onto D. I tried to solve this question in a lot of ways, for example, if we…
Tomás
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A caracterization of convexity in $\mathbb R^n$

Let $C$ be a closed subset of $\mathbb R^n$ such that $$\forall x,y \in C, (x,y)\cap C \neq \emptyset$$ where $(x,y)=\{(1-t)x+ty, t\in (0,1)\}$ Prove that C is convex A quick drawing shows that a concave set cannot satisfy this property (take $2$…
Gabriel Romon
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A convex real function is continuous - can we generalize?

There is a well-known theorem $\newcommand{\RR}{\mathbb{R}}$ Let $f: (a,b) \rightarrow \RR$ be a convex function. Then $f$ is continuous on $(a,b)$ The proof I know makes use of the fact that the real numbers form an ordered field and uses the…
marmistrz
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The join of two convex sets is convex?

Let A and B be convex subsets of $\Bbb R^n$. The join of A and B is the set of all $\vec x$ such that $\vec x$ lies on a line segment with one endpoint in A and the other in B. I am wondering how to show that the join of A and B is a convex set.
Allitee
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open convex cone coincides with the interior of its closure

Let $V$ be a finite dimensional real Euclidean space and $C$ be an open convex cone in $V$. I need to prove that $C = int(\overline{C})$. I proved that $C \subseteq int(\overline{C})$. I have difficulty in proving $int(\overline{C}) \subseteq C$. I…
La Rias
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