query: Are all open balls in $\mathbb R^2$ with respect to any metric convex?
positive_passages.docid: 147287
positive_passages.text: Is it possible to construct a metric in $\mathbb{R}^n $ s.t. it does not induce CONVEX balls?
negative_passages.docid: 43535
negative_passages.text: show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$
negative_passages.docid: 35573
negative_passages.text: A closed ball in a metric space is a closed set
negative_passages.docid: 9451
negative_passages.text: Show bounded and convex function on $\mathbb R$ is constant
negative_passages.docid: 57514
negative_passages.text: The Minkowski sum of two convex sets is convex
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: 15365
negative_passages.text: Proving that the triangle inequality holds for a metric on $\mathbb{C}$
negative_passages.docid: 72659
negative_passages.text: How can we define convexity in one dimension?
negative_passages.docid: 157753
negative_passages.text: Show that any set in a metric space can be written as the intersection of open sets
negative_passages.docid: 25017
negative_passages.text: Why is every $p$-norm convex?
negative_passages.docid: 149971
negative_passages.text: Closed unit disk homeomorphic to $\mathbb{R}^2$?
negative_passages.docid: 73927
negative_passages.text: Is $\mathbb{R}$ a subset of $\mathbb{R}^2$?
negative_passages.docid: 140692
negative_passages.text: Existence of a complete metric on $(0,1)$
negative_passages.docid: 6335
negative_passages.text: Proving that the set enclosed by an ellipse is convex
negative_passages.docid: 16943
negative_passages.text: Midpoint-Convex and Continuous Implies Convex
negative_passages.docid: 128894
negative_passages.text: About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$
negative_passages.docid: 77716
negative_passages.text: Not every metric is induced from a norm