query: Show that $f$ is convex iff $f\left (\frac{1}{2}x + \frac{1}{2}y \right ) \leq \frac{1}{2} f(x) + \frac{1}{2} f(y)$
positive_passages.docid: 16943
positive_passages.text: Midpoint-Convex and Continuous Implies Convex
negative_passages.docid: 60887
negative_passages.text: $d_A(x) = \inf_{a \in A} \|x-a\|$ is convex if and only if $A$ is convex
negative_passages.docid: 64955
negative_passages.text: Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
negative_passages.docid: 45742
negative_passages.text: Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
negative_passages.docid: 97562
negative_passages.text: Proving Lipschitz gradient of $f(x)=\sqrt{1+\|x\|^2}$
negative_passages.docid: 87207
negative_passages.text: Prove that the support function of an arbitrary set $A\subset\mathbb R^n$ is convex
negative_passages.docid: 35857
negative_passages.text: Characterization of log-convexity
negative_passages.docid: 52858
negative_passages.text: Find $f$ if $ f(x)+f\left(\frac{1}{1-x}\right)=x $
negative_passages.docid: 97584
negative_passages.text: How to prove this property using convexity?
negative_passages.docid: 144001
negative_passages.text: 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
negative_passages.docid: 121548
negative_passages.text: proof of an inequality $ \left( 1+ \frac{x}{p} \right) ^{p}$
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 116548
negative_passages.text: How to prove the real value function $f$ is convex on $[a,d]$ if $f$ is convex on both $[a,c]$ and $[b,d]$? ($a<b<c<d$).
negative_passages.docid: 21921
negative_passages.text: Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$
negative_passages.docid: 153191
negative_passages.text: A function satisfy $x\frac{\partial f(x,y)}{\partial x}+y\frac{\partial f(x,y)}{\partial y}=0$ in a convex domain implies it is a constant
negative_passages.docid: 7825
negative_passages.text: If $f$ is a concave function and $f(0) \geq 0$ then $f(a+b) \leq f(a) + f(b)$ for all $a,b >0$
negative_passages.docid: 40905
negative_passages.text: 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