MathDB
Lower bound on injectivity radius of graph

Source: Alibaba Global Math Competition 2021, Problem 14

July 4, 2021
metric spaceinjectivitygeometryanalysiscollege contests

Problem Statement

Let ff be a smooth function on Rn\mathbb{R}^n, denote by Gf={(x,f(x))∈Rn+1:x∈Rn}G_f=\{(x,f(x)) \in \mathbb{R}^{n+1}: x \in \mathbb{R}^n\}. Let gg be the restriction of the Euclidean metric on GfG_f.
(1) Prove that gg is a complete metric.
(2) If there exists Λ>0\Lambda>0, such that −ΛIn≤Hess(f)≤ΛIn-\Lambda I_n \le \text{Hess}(f) \le \Lambda I_n, where InI_n is the unit matrix of order nn, and Hess8f)\text{Hess}8f) is the Hessian matrix of ff, then the injectivity radius of (Gf,g)(G_f,g) is at least π2Λ\frac{\pi}{2\Lambda}.
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