query: Prove that $f$ is convex if and only if $f\left(\frac{a+b}{2}\right)\leq \frac{1}{b-a}\int_a^b f(x) dx$
positive_passages.docid: 150727
positive_passages.text: Characterization convex function.
negative_passages.docid: 45742
negative_passages.text: Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
negative_passages.docid: 152053
negative_passages.text: Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
negative_passages.docid: 34398
negative_passages.text: Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$
negative_passages.docid: 121548
negative_passages.text: proof of an inequality $ \left( 1+ \frac{x}{p} \right) ^{p}$
negative_passages.docid: 2848
negative_passages.text: Convex integral inequality
negative_passages.docid: 125175
negative_passages.text: convexity proof of a function including ln and sums
negative_passages.docid: 145884
negative_passages.text: Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
negative_passages.docid: 40438
negative_passages.text: Proving that closed (and open) balls are convex
negative_passages.docid: 69656
negative_passages.text: Prove or disprove that $\int_a^b |f(x)|\ \mathrm{d}x\geq \big | \int_a^b f(x)\ \mathrm{d}x\big| $
negative_passages.docid: 56081
negative_passages.text: Prove that the ellipsoid $E = \{ (x,y,z) \in R^3 \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1 \}$ 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: 15491
negative_passages.text: Prove that $f:\mathbb R^n\to\mathbb R$ is affine if and only if it is convex and concave
negative_passages.docid: 57514
negative_passages.text: The Minkowski sum of two convex sets is convex
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: 158399
negative_passages.text: Show that $\{(x_1,x_2):x_1x_2\geq4,x_1>0,x_2>0\}$ is a convex set