# Tagged Questions

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### 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 ...
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Consider the optimization problem $$(P) \quad \inf\{ f(x) : g_i(x) \le 0, i = 1,\ldots, m, x \in \mathbb R^m \}$$ where $f, g_i : \mathbb R^n \to (-\infty, \infty]$ are convex and $0 \ne ... 1answer 31 views ### Sufficient condition on convex function such that$f(x) > -\infty$for all$x$. Let$f : \mathbb R^n \to [-\infty, \infty]$convex and let$f(\overline x) > -\infty$for$\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that$f(x) > -\infty$for all$x \in \mathbb R$. ... 0answers 21 views ### Polar set and adjoint operator I'm trying to understand the following statement: A bounded operator between Banach spaces$u:X\rightarrow Y$satisfies: $$\textrm{inf}\{\;||u(B_X \cap S)||:S\subset X,\,\textrm{codim}S=k \; ... 0answers 22 views ### Dual convex pairs I am currently trying to understand a certain proof. The author uses the term dual convex pair for a pair (\phi,\psi) of convex functions defined on subsets X,Y of \mathbb R^n satisfying:$$ ... 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 ... 0answers 24 views ### What Projections preserve Pseudocontractiveness? Let$f: \mathbb{R}^n \rightarrow \mathbb{R}^n$be a pseudocontraction, i.e., $$\left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2$$ for all$x,y \in ...
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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 ...
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### Hyperplane which can separate two closed convex set

Define: $$E=:\ell^1(R)=\{x=(x_n)_n: \|x\|_1=\sum_{n=1}^{\infty}|x_n|<\infty\}$$ We have two closed convex sets $X,Y$ as subsets of normed vector space $\ell^1(R)$ with $X\cap Y=\emptyset$ and $Y$ ...
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### Relation between convex set and convex function

Let $E$ be an normed vector space and $A\subset E$ be a closed nonempty set. Define $$\phi(x)=\operatorname{dist}(x,A)=\inf_{a\in A}\|x-a\|$$ Prove that if $\phi$ is convex then $A$ is convex. ...
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### when convex diverging functions are monotone when divided by $x$

here is a calculus question that someone asked me to help him wuth and I have no answer for him. any help or ideas? Given $f:(0,\infty) \rightarrow \mathbb{R}$ is a convex function, and ...
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### Conjugate convex functions property

$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$ And I ...
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### Does $\phi\circ g$ convex imply $\phi\circ f\circ g$ convex?

Assume that $\phi$ is an increasing function and $g$ is a decreasing function with $\phi\circ g$ convex. If $f$ is an increasing convex function does this imply $\phi\circ f\circ g$ is a convex ...
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### Proving $x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$

Suppose $f$ is convex on $I$ and $(x,y,z)\in I^3$: How to prove that: $$x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$$
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### Can one define the derivative of a function using tangent cones? Does such a notion already exist?

I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
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### Pointed Convex cone: one-to-one correspondence extreme rays - extreme points

Hoi, let $V$ be a finite dimensional real vector space with inner product $\left\langle .\right\rangle$. Let $\Gamma\subset V$ and $\Gamma \neq \left\{0\right\}$ a pointed convex cone. (Pointed means ...
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### lsc function on compact set it attains its maximum minimum?

Is this true if so how to show it? if not true can you give a counter example: A lower semicontinuous function f on a compact set K attaings its minimum on K. A lower semicontinuous function f on a ...
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### Is this function convex

Is function $$f(x, y) = \left(\frac{x}{y} - a\right)^2 \left(\frac{y}{x} - \frac{1}{a}\right)^2$$ convex on the domain $$\{(x,y): x, y \in \mathbb{R}, x >0, y >0 \}\quad?$$ Now I think that it ...
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### Directions of decrease for a convex functions

Suppose $f(x,y)$ is a convex function and $$f(x+\Delta x, y) < f(x,y), ~~~ f (x, y + \Delta y) < f(x,y)$$ Does this imply $$f(x+\Delta x, y + \Delta y) < f(x,y)$$? I am guessing the ...
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### Weak convexity and continuity

For any open interval $(a, b)\subset {\mathbb R}\,$, define a weakly convex function $f:(a, b) \rightarrow {\mathbb R}$ as one for which $$f(q\;x_0 + (1 - q)\;x_1) \leq q\;f(x_0) + (1-q)\;f(x_1)$$ ...
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### convex conjugate $f^*$ is proper if both $f$ and $f^{**}$ are

If $f$ and $f^{**}$ on $\mathbb R^d$ are proper functions where $f^*$ stands for the convex conjugate of $f$ why does that follow that $f^*$ is proper, too? Thanks a lot...
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### affine function definition

If we define the affine function as $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for every $x,y \in R^d$ and $\lambda \in R$ How to show that it is equivalent to the definition ...
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### Simple proof? If $x$ lies outside a compact convex set, there exists a $y$ closer to every point in the set than $x$.

This seems rather obvious intuitively, but I can't find a simple proof. If $C$ is a compact, convex subset of $\mathbb{R}^n$ and $x \not \in C$, then there exists a point $y$ such that, for every ...
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### Jensen's inequality and a estimate in $L^p$

In problem 3 we have: If $f:\mathbb{R} \longrightarrow\mathbb{R}$ is mensurable, $E:=\mathrm{supp}\ f$ and $$\int_E e^{|f(x)|}dx =1,$$ then $f\in L^p(\mathbb{R})$, for all $p\in(0,\infty)$ and ...
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### How to prove this sign of the derivative?

Suppose that $u:[0,\delta]\rightarrow\mathbf{R}$, $u\in C^2((0,\delta))\cap C([0,\delta])$ such that $$u(0)=0,$$ $$u>0 \ \ in \ (0,\delta],$$$$u''>0$$ Then $u'>0$ in $(0,\delta)$.
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### Show this function is convex.

Could someone point me in the right direction for proving the following? Given that $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is an affine map given by $f(x)=A\mathbf{x}+\mathbf{b}$, ...
Let $C$ be a convex subset of a vector space $X$. A point $x\in C$ is called an extreme point if and only if whenever $x=ty+(1−t)z$, $t\in (0,1)$, implies $x=y=z$. It is known that the boundary ...