# Questions tagged [convex-hulls]

For questions on the convex hull of a set, a set $X$ of points in a Euclidean space which is the smallest convex set that contains $X$. Consider adding (convex-analysis), or, for questions related to algorithms, (computational-geometry) and/or (discrete-geometry).

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### Suppose $P_1$ and $P_2$ are two $n$-dimensional convex polytopes. Does $\partial P_1 \subseteq\partial P_2$ imply that $P_1 = P_2$?

Given two convex polytopes $P_1$ and $P_2$ with the same dimension, I want to know if the boundary of $P_1$ (denoted $\partial P_1$) being contained in the boundary of $P_2$ (denoted $\partial P_2$) ...
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### If $\mu$ is a probability measure, then $\int_{X}\phi \ \mathrm{d}\mu\in\mathbb{C}$ lies in the closed convex hull of $\phi(X)\subset\mathbb{C}$

Let $\mu$ be a probability measure on a measurable space $X$. Suppose that $\phi\colon X\to\mathbb{C}$ is integrable with respect to $\mu$. How does one prove that \int_{X}\phi \ \mathrm{d}\mu\in\...
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I have a discrete random variable $X$ and I know its expectation $\bar{X} = \mathbb{E}[X] \in \mathbb{R}$. I would like to say that the minimum of the support of $X$ must be $\leq \bar{X}$. Assuming ...