Puzzles

In progress.

1) Let $A \in \mathbb{R}^{n \times n}$ with $A_{ij} = \min(i, j)$. Show that the minimum eigenvalue of $A$ is at least $\frac{1}{4}$.

2) Let $X_1, X_2, \cdots, X_n$ be random variables such that any two have correlation at most $-\alpha$ for some $\alpha > 0$. What is the largest possible value of $n$?

3) Let $x_1, \cdots, x_n \in \mathbb{R}^d$ be iid and sampled from a standard Gaussian distribution. When is $0$ contained in their convex hull with high probability?

4) Let $X$ be a metric space such that every continuous function $f: X \to \mathbb{R}$ attains its maximum on $X$. Is $X$ compact?