# 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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### 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 ...
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### Comparison between secant and derivative in a convex function

Imagine that we have a function $f:\mathbb{R}\to\mathbb{R}$ which is convex, that is $f''>0$. We also know that $f'''<0$, that is its first derivative function is concave. Now, we can define its ...
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### Properties of convex function with Lipshitz continuous gradient (Prof. Nesterov's textbook)

I am reading the Prof. Nesterov's textbook: Introductory lectures on convex optimization - a basic course p.57 I have problem in the following: My ...
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### If $f$ is log-convex then $f$ is convex

Here's my attempt: $f$ is log-convex. Then: $\log f(\lambda x + (1-\lambda)y )\leq \lambda \log f(x) + (1-\lambda) \log f(y)$ As $e^x$ is increasing, we can apply it to the inequation without ...
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### Quasiconvexity (in the sense of Morrey) implies Rank-One convexity

I am trying to understand why Quasiconvexity implies Rank-One convexity. In a standard proof of this fact a sequence of functions is constructed, which converges weakly to zero in $W^{1,p}.$ in ...
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### How to prove a two variable set is convex

$X=\{(x,y)\in R^2\ :\ 3\le 2x+3y\le 8\}$ i tried to solve it as: Let set $X$ is convex for $x_2,y_2\in X$ such that $\alpha x_1+(1-\alpha)x_2$,$\alpha y_1+(1-\alpha)y_2\in X$ Now, $3\le 2x+3y\le 8$ ...
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### Proof of unique solution of strongly convex function (Prof. Nesterov Paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems I am confused about the green part of the following: $\\$ ...
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### How to prove this function is quasi-concave? [closed]

Consider the function $f(x,y) = x(1-y)\log(1+y/x^2)$, where $0\le x, y\le1$. Is this function quasi-concave?
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### Generalization of Brouwer’s fixed-point theorem

Perhaps the most widely known version of Brouwer’s famous fixed-point theorem reads as follows: For any $n\in\mathbb N$, let $A\subseteq\mathbb R^n$ be a compact (with respect to the Euclidean ...
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### Open ball around a point in the hyperplane in a locally convex space

Let $f$ be a continuous linear functional in a locally convex space $X$ and let $z\in X$ so that $f(z)=\alpha$. Let $V_z$ be an open neighborhood around $z$. It is my intuition that $V_z$ should ...
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### Probability that a point lies in an uncertain convex hull

Given $n+1$ independent random vectors $X_i \sim N(\mu_i,\Sigma_i)$, where each $\mu_i \in \mathbb{R}^n,$ let $C$ denote the random region formed by taking the convex hull of a realization of the set ...
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### Can we solve minimax in this way?

I am working to use proximal operators for solving a minimax optimization problem. It is known that if you use alternative optimization, the algorithm cycles, see an answer to this question convex-...
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### Minkowski sum and vectors

Problem: Given two convex polygons A, B, we can define Minkowski sum, as A + B = {a + b: a $\in$ A, b $\in$ B}, where a + b vector sum. Prove that: for every external perpendicular u to an edge of A,...
### Which is bigger $e^{(a+b)}$ vs $e^a + e^b$?
I understand that exponential function is a convex function so for any convex function $\theta(a+b) > \theta(a) + \theta(b)$, but can someone provide a more formal proofs ?