Convex functions in probability: $h(\alpha x_1 + (1 - \alpha)x_2) \le \alpha h(x_1) + (1 - \alpha)h(x_2)$
73074
$f \text{ convex }\Rightarrow \forall (x_0,f(x_0))\in \Gamma(f) \exists a : \forall x\in \mathbb{R}, f(x) \ge a(x-x_0)+f(x_0)$
27564
Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
122300
Maximizing a quadratic function subject to $\| x \|_2 \le 1$
36374
Exercise 1.6.3 from Alon & Spencer's *The Probabilistic Method*: prove that $Pr[|X-Y| \leq 2] \leq 3 Pr[|X-Y| \leq 1]$ for i.i.d. real RVs $X$ and $Y$
30386
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
153646
Convex combination of cumulative distribution functions
110222
Maximal inequality for a sequence of partial sums of independent random variables
46760
To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
144001
If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
143936
Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$
91500
Let $U, V \sim U(0,1)$ be independent. What is $P(U \leq V)$?
8963
Function that is both midpoint convex and concave
25017
Why is every $p$-norm convex?
23119
An application of Jensen's Inequality
49783
A possible inequality related to binomial theorem (or, convex/concave functions)
92029
Is this rational function convex?
35857
Characterization of log-convexity