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Found problems: 2

2021 Alibaba Global Math Competition, 14

Let $f$ be a smooth function on $\mathbb{R}^n$, denote by $G_f=\{(x,f(x)) \in \mathbb{R}^{n+1}: x \in \mathbb{R}^n\}$. Let $g$ be the restriction of the Euclidean metric on $G_f$. (1) Prove that $g$ is a complete metric. (2) If there exists $\Lambda>0$, such that $-\Lambda I_n \le \text{Hess}(f) \le \Lambda I_n$, where $I_n$ is the unit matrix of order $n$, and $\text{Hess}8f)$ is the Hessian matrix of $f$, then the injectivity radius of $(G_f,g)$ is at least $\frac{\pi}{2\Lambda}$.

2023 Miklós Schweitzer, 3

Tags: inequalities , metric space

Let $X =\{x_0, x_1,\ldots , x_n\}$ be the basis set of a finite metric space, where the points are inductively listed such that $x_k$ maximizes the product of the distances from the points $\{x_0, x_1,\ldots , x_{k-1}\}$ for each $1\leqslant k\leqslant n.$ Prove that if for each $x\in X$ we let $\Pi_x$ be the product of the distances from $x{}$ to every other point, then $\Pi_{x_n}\leqslant 2^{n-1}\Pi_x$ for any $x\in X.$