# 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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### Strong Convexity Inequality with Minimizer

I was reading "Primal-dual subgradient methods for convex problems" by Nesterov, and in the appendix, he proves that if $d(x)$ is $\sigma$-strongly convex, then it has a minimizer $x'$, and ...
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### Can a point be closer to all the vertices of a convex polytope than another point inside that polytope?

Consider a set $X = \{x_i \in \mathbb{R}^n\}$ and denote its convex hull $$C \equiv \bigg\{ \sum_i \lambda_i x_i : \lambda_i \geq 0 \text{ for all } i \text{ and } \sum_i \lambda_i = 1 \bigg\}.$$ I ...
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### Simple question on the gauge function

Suppose that $X$ is a real vector space and $K$ is a convex subset of $X$ that contains the origin as an interior point of $K$, i.e., if $y\in X$, then there exists an $\epsilon_{y} >0$ such that \...
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### How to see $\ell_0$-penalty is non-convex function?
$$\hat{\beta}(\lambda) = \arg \min_\beta (|| Y - X\beta ||_2^2 /n + \lambda ||\beta||_0)$$ where the $\ell_0$-penalty is $|| \beta||_0 = \Sigma_{j=1}^p 1(\beta_j \neq 0)$. The textbook says "...