Show that $f$ is convex in $I$ such that $f(\frac{x+y}{2})<\frac{1}{2}(f(x)+f(y))$
16943
Midpoint-Convex and Continuous Implies Convex
142360
How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
77217
Proof that a coordinate-wise convex function is convex?
97562
Proving Lipschitz gradient of $f(x)=\sqrt{1+\|x\|^2}$
71619
Proving that a convex function is Lipschitz
106718
Prove if C is midpoint convex and closed then its a convex set
64197
Subgradients of non-convex functions
92029
Is this rational function convex?
20614
Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex
61799
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
25738
Jensen's Inequality implies convexity?
138931
Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
597
f is differentiable on convex domain D and Re(f')>0 implies f is injective on D
74608
Convex function as supremum of countably many affine functions
73074
$f \text{ convex }\Rightarrow \forall (x_0,f(x_0))\in \Gamma(f) \exists a : \forall x\in \mathbb{R}, f(x) \ge a(x-x_0)+f(x_0)$
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
142922
How to prove that $e^x$ is convex?