# 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.

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### I’m studying engineering but I’d like to get more involved in pure mathematics.

I now realized I don’t like engineering that much and I love maths and I’m good at I. I want some book recommendations to get involved more in mathematics (especially convexity and fuzzy logic)
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### Supremum of a strictly convex function is always infinity?

Suppose there is a strictly convex continuous function $f$: $R^n$ $\rightarrow$ $R$. Is the supremum of $f$ always infinity? Does there exist any bounded strictly convex function?
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### Quadratic form can be represented as a convex combination of $\frac{n(n+1)}{2}+1$ ones

Question : In $\mathbb{R}^n$, consider an inner product $(\ ,\ )$. Here any linear map $L$ has the form $L(x)=(V,\ x)$ for some $V$ When $\|\ \|,\ \|\ \|^\ast$ are norms in dual relation, then we have ...
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### Minimizing a non-convex function through its component-wise convex functions

Let $f(x,y)$ be a continuously differentiable function from $\mathbb{R}^2$ to $\mathbb{R}$. I do not know whether $f$ is convex. But I do know that for any fixed $x$, $g_x(y)=f(x,y)$ is strictly ...
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### Higher order Jensen-like expansion upper bound

If $Z$ is a random variable with fine moment generating function, what is a good way to upper bound $$|\log \mathbb{E}e^Z- \mathbb{E}Z- \frac{1}{2}\mathbb{E}Z^2|$$ This looks like a third offer Taylor ...
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### How to prove the inverse image under an affine function is convex, if the image is convex?

Theorem in section 2.3.2 of Boyd & Vandenberghe's Convex Optimization: If $f:R^k \to R^n$ is an affine function and an set $S \subseteq R^n$ is convex, the inverse image of $S$ under $f$ defined ...
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### Duality in deterministic stochastic control and convex conjugate

I am currently reading the book "Stochastic Multi-stage Optimization" and trying to solve the Stochastic Optimal Control problem given in this book in the framework of duality. The problem ...
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### component Lipschitz constant

We had the following definition in class: Definition Suppose $f: \mathbb{R}^{n} \to \mathbb{R}$ continuously differentiable and $\nabla f(x)$ Lipschitz-continuously with constant $L > 0$, i.e. \...
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### Looking for a function that preserves concavity

Consider the function $$f(x,y) = g(x) y$$ where $g$ is some other function. We can restrict ourselves to $y\geq 0$ and $0\leq x\leq 1$. I would like to find a function $g$ with the following ...
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### Prove convexity (product of two functions)

Is there a way to prove convexity of $$f\left(\frac{x^{\mathsf T}\sigma}{\sqrt{x^{\mathsf T}\Sigma x }}\right)\sqrt{x^{\mathsf T}\Sigma x}$$ where $\sigma$ is a vector, $\Sigma$ is positive definite, ...
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Let $\phi$ be the Gaussian CDF, $\phi(t)=\int_{-\infty}^t\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \, dx$. We know that $\phi$ is log-concave. If we consider a combination of $\phi$ with a shift of $\... 1answer 36 views ### Does midpoint convexity at a single point imply full convexity at that point? Let$f:[a,b] \to \mathbb [0,\infty)$be a continuous function, and let$c \in (a,b)$be a fixed point. Suppose that$f$is midpoint-convex at the point$c$, i.e. $$f((x+y)/2) \le (f(x) + f(y))/2,$$ ... 1answer 22 views ### The converse of some theorems about Minkowski functional (From Lax's functional analysis) Let p denote a positive homogeneous subadditive function defined on a linear space$X$over the reals. (1)The set of points$x$satisfying$p(x) < 1$is a convex ... 3answers 46 views ### Showing a function is convex/concave given another function is concave Assuming$f(x)$is not a linear function, given a concave decreasing function$f(x)$, I want to find whether$g(x)=f(x)\cdot x$is convex or concave for strictly positive$x$. However, I'm having a ... 1answer 67 views ### Biggest convex set inside a concave unit ball Denote the unit ball for the$p$-norm in$\mathbb{R}^N$with$p \in (0,1]$, $$S_p^N = \Big \{ x \in \mathbb{R}^N,\ \Big(\sum \limits_{i=1}^N |x_i|^p\Big)^{1/p} \le 1 \Big\}$$ We want to find a convex ... 1answer 25 views ### Convex functions and minimizations [closed] triyin to solve this exercise, any advices and hints on how to prove it are more than helpful. The problem Let$C \subset \mathbb{R}^n$be a convex set, and$f: C \rightarrow \mathbb{R}$is also ... 2answers 22 views ### Is the function convex? Let's have the following function$f:\mathbb{R}^{2}\to\mathbb{R}$defined by$|x+y|$, is it convex? We have$\lambda\in (0,1),x,y\in$dom$(f)$, so$|\lambda x+(1-\lambda)y|\le \lambda |x| + (1-\lambda)...
Let $a, x \in \mathbb{R}$, $S \subset \mathbb{R}^n$ a convex and closed set and $L = S + \{x\}$. Prove that $L$ is convex and closed and that $proj_L(a) = x + proj_S(a - x)$. Guys, im stuck with this ...
Let $S \subset \mathbb{R}^n$ be a convex and closed set. Show that, given $x, y \in \mathbb{R}^n$: $$\|\operatorname{proj}_S(x) - \operatorname{proj}_S(y)\| \leq \|x - y\|$$ This questions seems to be ...