Intereting Posts

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In an additive category, why is finite products the same as finite coproducts?
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Measure Spaces: Uniform & Integral Convergence
Intuition behind Snake Lemma
How to find the vertex of a 'non-standard' parabola? $ 9x^2-24xy+16y^2-20x-15y-60=0 $
Is the derivative of a modular function a modular function
Holomorphic functions on a complex compact manifold are only constants
For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$
Does the little-oh relation remain if $f(x)$ and $g(x)$ both integrate or differentiate?
Number to the exponent divided by exponent value
Finding the limit $\lim_{n\rightarrow\infty}(1-\frac{1}{3})^2(1-\frac{1}{6})^2(1-\frac{1}{10})^2…(1-\frac{1}{n(n+1)/2})^2$
How big is infinity?
definit integral of Airy function

Let $\xi$ be a random variable supported in some set $A \in \mathbb{R}^n$: $\xi \in A$ a.e. How to show that $\mathsf E \xi \in \mathop{\mathrm{conv}}{A}$?

Let $s(x)$ be a support function of set $A$: $s(x) = \sup\limits_{y \in A} \langle x,y\rangle$, $x \in \mathbb{R}^n$. Then

$$y \in \mathop{\overline{\mathrm{conv}}}A \Leftrightarrow \langle x,y \rangle \leqslant s(x) \, \forall x$$

So $\langle x, \mathsf E \xi \rangle = \mathsf E \langle x, \xi\rangle \leqslant \mathsf E s(x) = s(x)$ since $\xi \in A \subseteq \mathop{\overline{\mathrm{conv}}}A$ a.e. This shows that $\mathsf E\xi \in \mathop{\overline{\mathrm{conv}}}A$. But how to show that we have stronger result: $\mathsf E \xi \in \mathop{\mathrm{conv}}{A}$?

- Proving Convexity of Multivariate Function using Conditional Univariate Convexity
- Show that the maximum of a set of convex functions is again convex
- About the strictly convexity of log-sum-exp function
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- subdifferential rule proof

- Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?
- Inequality with monotone functions on power set
- Finding a bound on the maximum of the absolute value of normal random variables
- Computing the characteristic function of a normal random vector
- A problem of regular distribution
- The problem of the most visited point.
- How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$
- Why are continuous functions not dense in $L^\infty$?
- How to prove this inequality about the arc-lenght of convex functions?
- Necessary and Sufficient Conditions for Random Variables

If $\mathbb E \zeta$ is on the boundary $\overline{\text{conv}} A\setminus \text{conv} A$, there is a hyperplane $H=\{x\mid \langle x,y\rangle=0\}$ containing $\mathbb E \zeta$, such that $A$ lies on one side, $\{x\mid \langle x,y\rangle\geq 0\}$. But $\langle \zeta,y\rangle$ is non-negative and has expectation zero, so must be zero almost surely: $\zeta\in H$ with probability $1$. The result then follows by induction on dimension and considering $A\cap H$.

Note that the result isn’t true in infinite dimensions: the (finitary) convex hull of unit vectors $(1,0,0,0,\dots),(0,1,0,0,\dots),\dots\in\mathbb R^{\infty}$ does not contain $(1/2,1/4,1/8,\dots)$ for example.

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