MathDB
IMC 2002 Problem 12

Source: IMC 2002

March 7, 2021
gradientinequalitiesreal analysis

Problem Statement

Let f:Rn→Rf:\mathbb{R}^{n}\rightarrow \mathbb{R} be a convex function whose gradient ∇f\nabla f exists at every point of Rn\mathbb{R}^{n} and satisfies the condition ∃L>0  ∀x1,x2∈Rn:    ∣∣∇f(x1)−∇f(x2)∣∣≤L∣∣x1−x2∣∣.\exists L>0\; \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||\leq L||x_{1}-x_{2}||. Prove that ∀x1,x2∈Rn:    ∣∣∇f(x1)−∇f(x2)∣∣2≤L⟨∇f(x1)−∇f(x2),x1−x2⟩. \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||^{2}\leq L\langle\nabla f(x_{1})-\nabla f(x_{2}), x_{1}-x_{2}\rangle.
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