Prove that $\varphi$ is convex if $\varphi \left[ \int_0^1 f(x) \, dx \right] \le \int_0^1 \varphi [f(x)] \, dx$ for any measurable $f$
130049
Converse of Jensen's inequality
67313
Proving $\{(x_1, x_2) \in \mathbb{R}^2_{+} \mid x_1 x_2 \geq 1\}$ is convex
129081
Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?
150727
Characterization convex function.
116548
How to prove the real value function $f$ is convex on $[a,d]$ if $f$ is convex on both $[a,c]$ and $[b,d]$? ($a<b<c<d$).
28007
Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $
138457
Seminorms: Absolute Convexity
143936
Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$
123732
Proving that the maximum of two convex functions is also convex
34398
Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$
10352
Prove that $x^n$ is a convex function on $(0,\infty)$
52231
Prove or disprove {$\sum_{i=1}^n x_i^2 = 1$} is convex
115814
If $\operatorname{Re}f^\prime > 0$ on a convex domain, then $f$ is one-to-one.
57943
If $G$ is a convex set, prove that $\mathbb{C}\cup\{\infty\}\setminus G$ is connected.
6335
Proving that the set enclosed by an ellipse is convex
20581
How to show that $\Gamma\vdash (\varphi\to\neg\varphi)\to\varphi$?
78229
Proving that the tangent to a convex function is always below the function