MathOverflow

Suppose we have an arbitrary Jordan curve $C$. Call $X$ the set of smooth approximations of $C$ within $\varepsilon$ of $C$. Is it true that there exists at least one $x\in X$ such that $x$ has an inscribed square with side length greater than $2\varepsilon...

Suppose I know that the non-negative integer-valued random variables $X_1,\dots,X_n$ satisfy that there exists $c>0$ such that for all $i\in [n]$ and $t\ge 0$$$\mathbb{P}(X_i\le \mathbb{E}X-t \mid X_1,\dots, X_{i-1})\le \exp(-ct^2).$$I am wonderingif...

Let $\mu$ be the Möbius function. It is well known that $\sum_{n|k} \mu(n) = 0$ for $k>1$. What could be said about the polynomials $R_k = \sum_{n|k} \mu(n) x^n$ for $x \in [0,1]$? There does not seam to an easy answer to that question. It could...

I found this evil exercise in chapter 1 of Engelking's and Sieklucki's book about topology, roughly:
Say $(X, Y)$ has non-expansive map property if for any $A\subseteq X$ and non-expansive $f:A\to Y$ there is non-expansive extension $F:X\to Y$.
Show the pairs 1) $(X, \mathbb{R}), 2) (\mathbb{R}^n, \mathbb{R}^n), 3) (S^n, S^n)$ have non-expansive map property....