query: If $X$ is finite dimensional, $A$ is a convex dense subset, how to prove that $\mathrm{Aff}(A)=X$?
positive_passages.docid: 154363
positive_passages.text: Prove that $\text{aff}(X) = \text{aff}(\text{closure}(X))$
negative_passages.docid: 130586
negative_passages.text: $f$ is discontinuous $\iff$ kernel($f$) is dense in X
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
negative_passages.docid: 73336
negative_passages.text: If $A$ is a nowhere dense subset of a topological space $X$, then $X\setminus A$ is dense in $X$.
negative_passages.docid: 147013
negative_passages.text: Subset of a finite set is finite
negative_passages.docid: 53132
negative_passages.text: A convex subset of a set has 'smaller' boundary than the set?
negative_passages.docid: 142922
negative_passages.text: How to prove that $e^x$ is convex?
negative_passages.docid: 71619
negative_passages.text: Proving that a convex function is Lipschitz
negative_passages.docid: 122463
negative_passages.text: How to show the that a set $A$ nowhere dense is equivalent to the complement of $A$ containing a dense open set?
negative_passages.docid: 54600
negative_passages.text: Is a convex and lower semicontinuous function defined on a closed and convex subset of $\mathbb{R}^n$ continuous?
negative_passages.docid: 7402
negative_passages.text: Showing a compact metric space has a countable dense subset
negative_passages.docid: 67313
negative_passages.text: Proving $\{(x_1, x_2) \in \mathbb{R}^2_{+} \mid x_1 x_2 \geq 1\}$ is convex
negative_passages.docid: 158582
negative_passages.text: Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
negative_passages.docid: 20789
negative_passages.text: If $\Sigma \vdash \varphi$ then $\Sigma \vdash \forall x \varphi$. Why when $(\varphi \implies \forall x \varphi)$ is not true?
negative_passages.docid: 72659
negative_passages.text: How can we define convexity in one dimension?
negative_passages.docid: 82824
negative_passages.text: Affine algebraic subset of $\mathbb{A}_k^4$
negative_passages.docid: 44430
negative_passages.text: Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$?