If $f$ is convex then $f\left(\sum_{i=1}^n\lambda_i x_i\right) \leq \sum_{i=1}^n\lambda_i f (x_i)$
40905
Prove $f(\sum^k_{i=1} \alpha_i x_i) \leq \sum^k_{i=1} \alpha_i f(x_i) $ for a convex function f
17336
Least norm in convex set in Banach space
106845
Why is log-of-sum-of-exponentials $f(x)=\log\left(\sum_{i=1}^n e^ {x_i}\right)$ a convex function for $x \in\mathbb R^n$?
14448
Elementary proof that $|x|^p$ is convex.
106718
Prove if C is midpoint convex and closed then its a convex set
40403
Show that the function is convex
85465
Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.
114575
If $\sum_{i=1}^k \dim F_i > n(k-1)$ then $\cap_{i=1}^k F_i \neq \{0\}$
123732
Proving that the maximum of two convex functions is also convex
67454
Show that $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^n (x_i - \bar{x})(y_i)$.
97584
How to prove this property using convexity?
35857
Characterization of log-convexity
40438
Proving that closed (and open) balls are convex
52231
Prove or disprove {$\sum_{i=1}^n x_i^2 = 1$} is convex
7825
If $f$ is a concave function and $f(0) \geq 0$ then $f(a+b) \leq f(a) + f(b)$ for all $a,b >0$
89429
Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
61162
Show that $\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x\, f(x)^2\,dx $