# Tagged Questions

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### Affine to linear like conversion of a concave function

Is the following true: $$\log \left( \frac{1}{f(x)+K}\right)\mathrm{is\;concave}\Longleftrightarrow \log \left( \frac{1}{f(x)}\right)\mathrm{is\;concave},$$ where $K\in\mathbb{R}$ and ...
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### Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
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### Convexity of functions

I'm trying to read up on convex/concave functions (is that the same as concave up, concave down?) If I were asked to prove a convexity of a function, what are the general steps to follow? (So far I ...
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### Composition of a convex function and a convex decreasing function is quasi-concave

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h(g(x))$ is a quasi-concave function?
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### How to check the function is convex or not?

How one can check the function whether it is convex or not. I know one method by using Hessian Matrix but I think it did not fit for the following example. I think Hessian matrix method cannot be ...
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### Question about strongly convexity and affinity

For a function $f$, it is said to be strongly convex if for all $x,y$ $$(\nabla f(x) - \nabla f(y) )^T (x-y) \ge m \|x-y\|_2^2$$ for a constant $m \ge 0$ Is it called ...
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### mid-point convex but not a.e. equal to a convex function

I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpniski's theorem from which we can deduce that for ...
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### Is $\sqrt{x}$ concave?

I have function $f(x)= \sqrt(x)$. To check is it concave or convex i am checkin $f''(x).$ Which is $-\frac{1}{4x^{\frac{3}{2}}} < 0$ So the $f(x)$ is concave. Is it correct ? And is is the same ...
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### Properties concave functions

Is is true that if $f(x)$ is a concave function of $x$ with domain $C$, then $f'(a) \leq \frac{f(a)}{a}$ for any $a \in C$, where $f'(a)$ denotes the derivative of $f(x)$ with respect to $x$ evaluated ...
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### How to prove that the product of a decreasing monotonic function and a strictly increasing monotonic function is a concave function?

Given that I have a strictly increasing monotonic function $f$ and a decreasing monotonic function $g$, are there any nice properties to show that the product function $h(x) = f(x)g(x)$ is a concave ...
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### Is the function of two strictly concave functions also concave?

This may be a trivial question to most, but here we go: I have two strictly concave functions, say $f(x)$ and $g(x)$. From this can I say that a function of those two functions, $h[f(x), g(x)]$, is ...
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### Why can we assume WLOG that $x$ is zero?

I am new to mathematics, so I apologize in advance if this question is trivial. I was trying to prove a property of an arbitrary three-point system in $\mathbb{R}^2$ regarding convexity. I tried it ...
480 views

### sum of concave and convex function

Suppose $f$ is the sum of a concave and convex function, i.e. $$f=f_1+f_2$$ where $f_1$ is a concave function and $f_2$ is a convex function. I wonder if $f$ can be written as the following: ...
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### Is the following function of several variables concave?

Suppose that $w_1,\dots,w_p\in(0,1)$ satisfy the condition $\sum_i w_i=1$, and let $$F(w_1,\dots,w_p) = \frac{1-\sum_i w_i^2}{\sum_i(1-w_i)^2 \left[\sum_i \frac{w_i}{1-w_i}\right]^2}.$$ Is function ...
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### Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
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### convex relaxations

Is there a notion of "best" convex relaxation of a particular function in a normed vector space? For example, the $\ell^0$ pseudo-norm can be relaxed into the $\ell^1$ norm, and that allows us to ...
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### A sequence of differentiable convex function converges uniformly to a convex function?

Let $f$ be a convex function. How to prove that there is a sequence of differentiable convex function that converges uniformly to $f$?
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### How to verify the following function is convex or not?

Consider function $$f(x)=\frac{x^{n_{1}}}{1-x}+\frac{(1-x)^{n_{2}}}{x},x\in(0,1)$$ where $n_{1}$ and $n_2$ are some fixed positive integers. My question: Is $f(x)$ convex for any fixed $n_1$ and ...
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### Intersection of two (related) concave functions

Question: In general, two concave functions intersect at at most two points. True or False? If false, can you please provide an example. If true, can you please provide a proof. Proving or disproving ...
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### Properties of nearly concave functions

Is there any existing literature on the properties/applications of the following class of functions? $$\frac{f(E[x])}{E[f(x)]}\geq c$$ where $c< 1$ is a constant. Note that for $c=1$ these are ...
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### Generating an N-Dimensional Convex Quadratic Function

I am currently working on a research project that is trying to show that some numerical integration techniques that do well on separable convex functions will not necessarily do well on non-separable ...
Consider a function $f$ over $\Re^n$ to $\Re$. Suppose it is true that for every affine subspace with dimension strictly lesser than $n$ the function $f$ is concave. Is the function $f$ concave over ...
I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave. There could be ...