query: Is $\phi$ convex if it is continuous and $\phi \Big(\frac{x+y}{2}\Big)\le \frac{1}{2}\phi (x) +\frac{1}{2} \phi (y)$
positive_passages.docid: 25738
positive_passages.text: Jensen's Inequality implies convexity?
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: 71619
negative_passages.text: Proving that a convex function is Lipschitz
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: 142922
negative_passages.text: How to prove that $e^x$ is convex?
negative_passages.docid: 21301
negative_passages.text: If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
negative_passages.docid: 151410
negative_passages.text: Inequality of continuous functions
negative_passages.docid: 142740
negative_passages.text: If $\varphi \in E'$ and $A$ is convex and open then $\varphi (A)$ is an open interval
negative_passages.docid: 94231
negative_passages.text: Prove that a non-empty closed convex $S$ has an extreme point iff it does not contain any line
negative_passages.docid: 133228
negative_passages.text: Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
negative_passages.docid: 40438
negative_passages.text: Proving that closed (and open) balls are convex
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: 32611
negative_passages.text: Is the trace of inverse matrix 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: 102959
negative_passages.text: Show that $f$ is convex if and only if $f\left( \sum_{i=1}^m\lambda_ix_i \right) \leq \sum_{i=1}^m\lambda_if(x_i)$
negative_passages.docid: 67253
negative_passages.text: Triangle inequality univalent
negative_passages.docid: 30423
negative_passages.text: interior points and convexity