# Questions tagged [convex-geometry]

Use this tag when posting questions related to the concept of convexity and geometry. For example, for convex polygons.

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### convex sets in convex optimization

How to prove that the following set is not convex? $$M = \left\{ \mathbb{R}^{3}: x_{1}x_{2}x_{3}\le 1,x_{1}+x_{3}\ge 2,x_{1} \ge 0 \right\}$$ Thanks for any help. I tried to write it down as ...
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### Do convex combinations of projection matrices majorize the probability vector, i.e. $\sum_k p_k P_k\succeq \boldsymbol p$?

Consider a convex combination of normal projection matrices with positive coefficients: $$C\equiv \sum_k p_k P_k,$$ where $p_k>0$, $\sum_k p_k=1$, and $P_k=P_k^\dagger=P_k^2$. If the $P_k$ are ...
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### Fixed area polygon maximize the area of its convex hull

For a fixed area, what kind of polygon that would obtain the maximum area of its convex hull. For e.g. area 1 square, since it’s already convex, so the convex hull having area 1 too. Would there be ...
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### Is Inverse Cartesian product of convex set still convex?

Given two compact convex set $X \subset \mathbf{R}^2, Y \subset \mathbf{R}^2$. The Cartesian product $Z := X \times Y \subset \mathbf{R}^4$ Is again a convex subset. Is the low dimension projection ...
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### 'Shrunken Version' of a convex set is also convex

I'm trying to show that for a convex set $K$ in $\mathbb{R}^n$ (possibly bounded, if that makes things easier), the set $K_{\epsilon}:= \{x\in K: \text{dist}(x,\partial K)>\epsilon\}$ is also ...
$\newcommand{\CO}{\text{CO}}$ $\newcommand{\SO}{\text{SO}}$ $\newcommand{\dist}{\text{dist}}$ Let $\CO(2) =\{\lambda R : R \in \SO(2)\, | \, \lambda > 0\}$ be the set of $2 \times 2$ conformal ...
I want to minimize function $\underset{T,\alpha }{\mathop{\min }}\,F\left( T,\alpha \right)$, where T is continue variable and $\alpha$ is a subset of $\beta$ with constant c members, i.e. \$\begin{...