query: Let p ∈ [1,+∞[. Show that f: X → R: x 􏰅→ ∥x∥^p is convex.
positive_passages.docid: 143936
positive_passages.text: Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$
negative_passages.docid: 88687
negative_passages.text: Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
negative_passages.docid: 61799
negative_passages.text: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
negative_passages.docid: 57514
negative_passages.text: The Minkowski sum of two convex sets is convex
negative_passages.docid: 46760
negative_passages.text: To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
negative_passages.docid: 60521
negative_passages.text: Continuity of a convex function
negative_passages.docid: 40438
negative_passages.text: Proving that closed (and open) balls are 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
negative_passages.docid: 31548
negative_passages.text: Proof of "every convex function is continuous"
negative_passages.docid: 46746
negative_passages.text: How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?
negative_passages.docid: 3273
negative_passages.text: Trouble in proving that $\|x\|_p = \max|x_j|$
negative_passages.docid: 7775
negative_passages.text: Show that $|x-y|\leq |x|+|y|$.
negative_passages.docid: 25738
negative_passages.text: Jensen's Inequality implies convexity?
negative_passages.docid: 15645
negative_passages.text: Is convex open set in $\mathbb{R}^n$ is regular?
negative_passages.docid: 137511
negative_passages.text: Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
negative_passages.docid: 119735
negative_passages.text: Show that $\mathbb{E}(g(T-p)) < \mathbb{E}(g(S-p))$ for any convex function $g$ if $T$ and $S$ are estimators of $p$
negative_passages.docid: 52231
negative_passages.text: Prove or disprove {$\sum_{i=1}^n x_i^2 = 1$} is convex