# Tagged Questions

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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### Let B be set of all twice differentialbe function $f(0)=1, f'(0)=-1$ . .. Find supremum of ${(f''(0):f\in B})$

Let B be set of all twice differentiable function $f$ such that $f: (-1,1) \to (0,\infty)$ and $f(0)=1, f'(0)=-1$ . We have new function $g(x)$ such that $g(x)=\frac{1}{f(x)}$ and $g(x)$ is convex ...
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### Some intuition on the support function of a convex set

I have some doubts on the interpretation and properties of the support function of a convex subset of $\mathbb{R}^d$. (1)Let $K$ be a convex set in $\mathbb{R}^d$. (2) The support function of $K$ is ...
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### Is $(\sum_{i=1}^n a_i{x_i}^2)(\sum_{i=1}^n b_i{x_i}^2)$ convex?

I was trying to solve a problem and at a point I needed that $$(\sum_{i=1}^n a_i{x_i}^2)(\sum_{i=1}^n b_i{x_i}^2),$$ $a_i,b_i >0$ is convex. So, I tried instead to prove that $x^2y^2$ is convex( ...
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### convex hull of a lower hemicontinuous correspondence is lower hemicontinuous

Let $(X, \mathcal{T})$ and $(Y, \mathcal{T}')$ be two topological spaces. We call a correspondence (set-valued function) $F: X \rightarrow 2^Y$ lower semicontinuous if for every $G \in \mathcal{T}'$ ...
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### Log-convexity of completely montone sequences

Let $s_0, s_1, \ldots$ be a completely monotone sequence. This means that, defining \begin{align*} (\nabla s)_n &= s_{n}-s_{n+1}\quad\text{and}\\ (\nabla^{r+1}s)_n &= (\nabla^{r}s)_n - (\...
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### $\big\langle\nabla f(x)-\nabla f(y),\,(x-y)\big\rangle\ge m\,\left\| x-y\right\|^2,\;m>0\;$ for strictly and strong convex function [closed]

Prove that $\,\big\langle\nabla f(x) - \nabla f(y),\, (x-y)\big\rangle \geq m\,\left\lVert x-y\right\rVert^2, \;\,m > 0\,$ for strong convex function $f: \mathbb R^n \to\mathbb R$. Is it true for ...
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### Proof of $A\ne \mathrm{conv}~ \mathrm{relbd} A\Rightarrow A$ is a flat or a half-space

Picture below is from the 16 page of Schneider R. "Convex Bodies--The Brunn-Minkowski Theory". I am fuzzy with the red line, what is the translate of $H^-$ ? And what is the effects of convexity ...
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Assume $f:\mathbb{R}^n \to \mathbb{R}$ is convex, and $L$-Lipschitz, so $|f(x)-f(y)|\leq L\|x-y\|$. I would like to show that $\|\nabla f(x)\|\leq L$. In one dimension this is a straightforward ...
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### Is there a way to determine if the Convex Hull of two polyhedra is going to be huge?

So in this post: Faster Algorithms for Convex Hulls I was interested in determining if a convex hull of two $n$ dimensional polyhedra can be computed quickly, and the answer was in general: no, ...
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### A consequence of the convexity of $f(x) = x \log x$

I verified that $f:\mathbb{R_{+}^{*}} \rightarrow \mathbb{R}, f(x) = x \log x$ is convex, since it is twice differentiable and $f''(x) = \frac{1}{x}$ is positive for the domain. But my teacher asked ...
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### Is $\Gamma(\alpha, 0, x)$ log-concave as a function of $x$ (for fixed $\alpha$)?

The incomplete Gamma function is defined as: $$\Gamma(\alpha, 0, x) = \int_0^x t^{\alpha - 1} e^{-t} dt$$ Let $\alpha \ge 1$ be a fixed number. Is $\Gamma(\alpha, 0, x)$ log-concave, as a function ...
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### Is a convex cone a convex polyhedron?

Say that I have a convex cone $C=\{t|Ax = t, x\geq 0\}$. where $x\in R^n$, and $t\in R^m$, $A\in R^m\times R^n$. Can I say that this is a convex polyhedron? and why? EDIT: Just in order to avoid ...