2
votes
0answers
46 views
+50

Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
0
votes
1answer
40 views

Proof of the Hahn-Banach separation theorem

In the proof of the Hahn-Banach separation theorem my notes claim the following: Let $X$ be a normed $\mathbf{R}$ vector space, $A,B\subset X$ be nonempty, disjoint, convex, $A$ compact and $B$ ...
1
vote
1answer
54 views

New variable in a convex optimization problem

Consider the convex optimization program $$ \min_{x \in X } x^\top P x + p^\top x \quad \text{ sub. to: } Ax = b $$ where $X \subset \mathbb{R}^n$ is compact, $P \succ 0$, $A \in \mathbb{R}^{m \times ...
0
votes
1answer
24 views

On level set of concave function

The problem is to show the following: Let $\varphi$ be a closed concave function, and $M=\max_{x \in \mathbb{R}^d} \varphi(x)$. Let $D_r:=\{\varphi\geq r\}$ be the level set. Then given $r \leq s ...
3
votes
1answer
59 views

How to prove Godunova's inequality?

Let $\phi$ be a positive and convex function on $(0,\infty)$. Then $$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$ The ...
0
votes
1answer
19 views

Question about relative interiors and convexity

Suppose that $C\subseteq \mathbb{R}^n$, such that $\operatorname{ri} C\neq \emptyset$ is convex and $\operatorname{cl} C$ is convex. Can we show that ...
4
votes
1answer
54 views

Let $S$ be a closed convex set & $x$ be an extreme pt of $S$ then $S-\{x\}$ is

Let $S$ be a closed convex set and $x$ be an extreme point of $S$, then $S-\{x\}$ is Convex Not Convex May or may not be convex I am thinking that the convexity doesn't fail even if we remove the ...
0
votes
2answers
22 views

Difference quotient inequality with convex functions

A convex function on an interval $ I $ is said to be convex if for every $ 0 < t < 1 $ and $ x,y \in I $ we have that $ f(tx + (1-t)y) \le tf(x) + (1-t)f(y)$. Prove a function is convex if and ...
1
vote
1answer
22 views

Supremum of convex lipschitz functions.

Let $f_i:K\to R, i\in I$ be a family of convex, equi-Lipschitz functions on some compact subset $K$ of $\Bbb R^n$. Is it true that $\sup f_i$ is also Lipschitz continuous(assuming that the sup ...
2
votes
0answers
21 views

Convexity in each argument and directional derivative

Let $f(x,y)$ be a continuous function, convex in each argument separately. Does this imply the existence of one-sided directional derivatives in any direction? For example, does there exist (finite or ...
0
votes
0answers
34 views

Sufficient conditions for convexity using the right derivative

We have a function $f:[0,1] \rightarrow \mathbb{R}$ that is continuous on $[0,1]$ with a non-decreasing right derivative everywhere in $(0,1)$. Is this definitely sufficient to show that $f$ is convex ...
0
votes
0answers
23 views

Dual convex pairs

I am currently trying to understand a certain proof. The author uses the term dual convex pair for a pair $(\phi,\psi)$ of convex functions defined on subsets $X,Y$ of $\mathbb R^n$ satisfying: $$ ...
2
votes
1answer
44 views

A question about norm for bounded linear transformations

Let $H$, $K$ be Banach spaces, and let $A: H \rightarrow K$ be a bounded linear transformation. Its norm is defined by: \begin{equation} \|A\| = sup\{\|Ah\|_K: \|h\|_H \le 1\} \end{equation} How to ...
2
votes
0answers
118 views

Convex subsets, Normed spaces, Separating hyperplane

I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then ...
0
votes
0answers
18 views

Robust LP question using box uncertainty model

I am trying to solve this robust LP problem by writing it as a QP $$\min_x x^TSx : \mu \leq r^T x , Ax \leq b$$ Under Box uncertainty model: $$R = \{r : \| r - \hat{r}\|_\infty \leq \rho\}$$ Here ...
1
vote
1answer
32 views

Proving upperbound using convexity

The original question is to prove $$\frac{1}{n}\sum_{i=1}^n x_i \leq \log{(\frac{1}{n}\sum_{i=1}^n e^{x_i})} \leq \max_{1 \leq i \leq} x_i$$ I show that $$x_{max} = \max_{1 \leq i \leq} x_i$$ ...
1
vote
1answer
19 views

Convexity and Jensen's Inequality for simple functions

Suppose $\varphi$ is convex on $(a,b)$. I want to show that for any $n$ points $x_1,\dots,x_n \in (a,b)$ and nonnegative numbers $\theta_1,\dots,\theta_n$ such that $\sum_{k=1}^n \theta_k = 1$ we are ...
0
votes
1answer
24 views

Decreasing Function Projected onto Simplex

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, defined as $f(x) := a x + b$, where $a<0$ and $b \in \mathbb{R}_{\leq 0}^{n}$. Note that $f$ is decreasing: $$ x \geq y \Longrightarrow f(x) ...
1
vote
2answers
51 views

Minimization of log-sum-exponential function subject to constraints.

I would like to minimize the following function: $f(x)=log(e^{-x_1}+..+e^{-x_n})$ Subject to: $\sum_{i=1}^{n}{x_i}=1$ $0 \leq x_i \leq 1$ So far I have discovered the following: If all the ...
0
votes
1answer
30 views

How to prove that $f$ is convex function

Let $f:(a,b) \mapsto \mathbb{R}$ be continuous function such that $f(\frac{x+y}{2})\leq \frac{1}{2}f(x) + \frac{1}{2}f(y) \;\; \forall x,y \in (a,b)$ Show that $f$ is convex function. Please give ...
0
votes
2answers
26 views

Notion of a concave function and proving ln is concave

I've just checked that the definition is right, a function is convex if: $(1-t)f(x_1)+tf(x_2)\ge f((1-t)x_1+tx_2)$ which is odd because this is ... well I was taught (very young age) that concave ...
0
votes
0answers
36 views

Minimization of product function subject to constraints

I want to minimize the following function: $\prod_{i=1}^{n}{x_i}$ Subject to the following constraints: $\sum_{i=1}^{n}{x_i}=1.1+(n-1)(0.1)$ and $0.1 \leq x_i \leq 1.1$ How should I go about it? ...
1
vote
1answer
35 views

Show that for $f,g: [0, + \infty) \to \mathbb{R}$ convex functions of Class $C^2$ their product $fg$ is convex

Problem: Let $f,g: [0, + \infty[ \to \mathbb{R}$ be two convex functions of Class $C^2$. Assume that $$ f(0) \geq 0, g(0) \geq 0 \text{ and } f'(0) \geq 0, g'(0) \geq 0 \tag{!}$$ Show that $fg$ ...
1
vote
1answer
28 views

convex closed and unclosed functions and (lower semi)continuity

I'm grappling a bit with lower semicontinuity and convex functions. Let me consider convex functions as functions to $\mathbb{R}$ and not to $\mathbb{R}\cup \{\pm\infty\}$. By Rockafellar's book ...
1
vote
1answer
27 views

Equality constrained Quadratic Program

Consider the QP $$ x^* = \arg \min_{\displaystyle x \in \mathbb{R}^n{\geq 0}} \ \frac{1}{2} x^\top P x + q^\top x \ \text{ sub. to: } A x = b, $$ where $P \succ 0$. Without the non-negativity ...
1
vote
1answer
67 views

Strictly convex unit balls in $L^p$

I need to show that if $1<p<\infty$, then the unit ball is strictly convex in $L^p$, that is, $||\lambda x+(1-\lambda)y|| < 1$ whenever $||f|| = ||g||=1$ and $\lambda \in (0,1)$. I tried ...
0
votes
1answer
20 views

On the Composition of simple Projections

Consider the compact convex set $X = \{ x \in \mathbb{R}^n \mid x \geq 0, \ \underline{1}^\top x = 1 \}$. I am wondering if the projection onto $X$ is the composition of the projection on $[0,1]^n$ ...
1
vote
1answer
37 views

Quadratic programs: is the projection onto constraints optimal?

Consider the Quadratic Program $$ x^* := \arg \min_{ x \in X } \ \{ x^\top x + c^\top x \} \ \text{ sub. to: } Ax=b $$ where $X \subset \mathbb{R}^n $ is a non-empty, convex, bounded polyhedron. ...
0
votes
3answers
62 views

A convex function has a lower bound?

Suppose that $f=f(x)$ is strictly convex for $x\in\mathbb{R}$, i.e. there exists $\epsilon>0$ such that $f''(x)\geq\epsilon>0$ for $x\in\mathbb{R}$. Does there exist $\delta>0$ such that ...
0
votes
1answer
25 views

Certain property of convex functions…

I come to you with yet another qualifying problem we can't seem to solve... Let $f:$ $(0,\infty) \to \Bbb R$ be convex, and let $\lim_{x \to 0}f(x)=0$. Show that $g(x)$ = $f(x) \over x$ is increasing ...
0
votes
1answer
21 views

Redefine a discrete compact set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
0
votes
1answer
23 views

Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
2
votes
2answers
49 views

Prove: $f: (a,b) \rightarrow \mathbb{R}$ is convex iff for every triple $x_1,x,x_2 \in (a,b)$…

Assignment: Prove that a function $f : (a,b) \rightarrow \mathbb{R}$ is convex if and only if for every triple $x_1,x,x_2 \in (a,b)$ with $x_1 < x < x_2$ following inequality is satisfied: ...
2
votes
1answer
48 views

Let $f: [a,b] \rightarrow R$ be convex. If f is not constant, then the supremum of $f$ is not inside of [a,b].

I could use some help with this proof. Let $a,b \in \mathbb{R}, \ a<b \ \ f: [a,b] \rightarrow \mathbb{R}$ be convex. Prove: If f is not constant, then the point, where the supremum of ...
0
votes
0answers
44 views

Creating a $C^2$ smooth convex curve which joins circular arcs

Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their ...
1
vote
1answer
14 views

Properties that guarantee quasiconvexity in $\mathbb{R}^n$

I have an open bounded connected domain $\Omega \subseteq \mathbb{R}^n$ and I would like to say that for every two $x,y \in \Omega$ there is a path $\gamma$ from $x$ to $y$ of length at most $C|x-y|$ ...
0
votes
1answer
30 views

Integral of increasing continuous function is convex

Suppose $g$ is increasing and continuous. Does it follow that $G(x) = \int_0^xg(y)dy$ is convex? Clearly $G'$ is increasing and continuous, and $G''\geq 0$ exists a.e., but I don't see how this ...
5
votes
1answer
116 views

A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
0
votes
0answers
22 views

Convex in $ \mathbb{R^n}$

Prove that: [A be a convexe part $(A\subseteq \mathbb{R^n})] \implies [\forall x_1,x_2,...x_n\in A ,\forall\alpha_1,\alpha_2,...\alpha_n\ge0 $ $with$ $ \ \alpha_1+\alpha_2+...+\alpha_n=1 ...
0
votes
1answer
24 views

Convex function almost surely differentiable.

If f: $\mathbb{R}^n \rightarrow \mathbb{R}$ is a convex function, i heard that f is almost everywhere differentiable. Is it true? I can't find a proof (n-dimentional). Thank you for any help
0
votes
2answers
68 views

Solution in general for a seemingly simple problem

Let $\mathbb{S}$ be a closed, bounded, convex set in $\mathbb{R}^N$. Let $\mathbb{x}=[x_1,\dots,x_N]$ be any arbitrary vector in $\mathbb{S}$. Then what can we comment on the problem \begin{align} ...
0
votes
1answer
50 views

How can the sum of two closed cones be not closed?

Can there be two closed cones $K_1$ and $K_2$ in $\mathbb{R}^3$ such that $K_1+K_2$ need not be closed?
0
votes
1answer
40 views

Intersection of strictly monotone, convex functions on interval with two given intersections

Given two functions $f,g:[0,1]\rightarrow[0,1]$ that are smooth, strictly convex, strictly monotone s.t. $f(1)=g(1)=0$, $g(0)=f(0)=1$ and $f(x_0)>g(x_0)$ holds for some point $x_0$, does it follow ...
1
vote
1answer
45 views

Sets defined by distance to a convex set

Let $Y \subset \mathbb R^n$ be a bounded convex set, let $R>0$, and let $$Z := \left\{z \in \mathbb R^n : d(z,Y) > \dfrac12R \right\}$$ where $$ d(z,Y) = \inf_{y\in Y}|y-z|. $$ If you like, ...
0
votes
0answers
26 views

Lipschitz in R^1 implies Lipschitz along any line in R^k (convex function)

I'm trying to solve a homework problem (baby rudin course) and realized that Lipschitz continuity really makes the problem easier for me. However, since it is not defined in Rudin I have to prove the ...
0
votes
2answers
35 views

Convexity of support function

Let $C$ be a closed non-empty set, but not necessarily convex. The support function of $C$ is given by $$S(z) = \sup_{c \in C} \langle z,c\rangle. $$ Prove that this is a convex function. ...
3
votes
0answers
82 views

Convex Functions are Continuous

In W. Rudin's Principles of Mathematical Analysis, we read in Chapter 4 that real-valued functions defined on an open interval $(a,b)\subseteq\Bbb R$ are continuous (specifically, Exercise 23). I am ...
0
votes
0answers
28 views

Show that the Rosenbrock function is strictly convex for a specific region

So we know that the Rosenbrock function is a test function of sorts, but can anyone prove that a specific region is strictly convex? Rosenbrock eqn: $ f(x_{1},x_{2}) = 100(x_{2} - x_{1}^{2})^{2} + ...
0
votes
1answer
88 views

Can a Lipschitz continuous function be strictly convex?

Let $\varphi:\mathbb R^n\to\mathbb R$, and suppose for all $x,y\in\mathbb R^n$, $$\|\varphi(x)-\varphi(y)\|\leq L\|x-y\|$$ for Lipschitz constant $L$. Is it possible for such a function to satisfy ...
1
vote
2answers
105 views

Tangent line of a convex function

Let $V : \mathbb{R}^n \rightarrow \mathbb{R}$ be (strictly) convex and continuously differentiable on $\mathbb{R}^n$. Show that for any $y \in \mathbb{R}^n$ and any $x \in \mathbb{R}^n$ $$ V(y) \geq ...