Define the distance function ds(): R" → R, and show that |ds(x) — ds(y)| ≤ ||x − y|| V x, y ER".
Define the distance function ds(): R" → R, and show that |ds(x) — ds(y)| ≤ ||x − y|| V x, y ER".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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This is from an optimization and convex analysis class
![**Problem 1**
Define the distance function \( d_S(\cdot) : \mathbb{R}^n \rightarrow \mathbb{R} \), and show that
\[
\left| d_S(\mathbf{x}) - d_S(\mathbf{y}) \right| \leq \left\| \mathbf{x} - \mathbf{y} \right\| \quad \forall \, \mathbf{x}, \mathbf{y} \in \mathbb{R}^n.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe87a569b-7721-4104-9b29-fedcf0a2817b%2Fde77b984-f5a4-4668-8f27-a3f8bb970e4d%2F8b7nj7s_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 1**
Define the distance function \( d_S(\cdot) : \mathbb{R}^n \rightarrow \mathbb{R} \), and show that
\[
\left| d_S(\mathbf{x}) - d_S(\mathbf{y}) \right| \leq \left\| \mathbf{x} - \mathbf{y} \right\| \quad \forall \, \mathbf{x}, \mathbf{y} \in \mathbb{R}^n.
\]
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