I recently watched this excellent video about Fourier transforms from one of my favorite Youtube channels -- 3Blue1Brown:
At the end of the video, Grant presents this problem:
\(\mathcal{C}\) is some convex subset of \(\mathbb{R}^3\).
\( \mathcal{D} = \{ \textbf{p} + \textbf{q} \mid \textbf{p}, \textbf{q} \in \partial \mathcal{C} \} \)
Prove that \( \mathcal{D} \) is convex.
Here, \( \partial \mathcal{C} \) denotes the boundary of the set \( \mathcal{C} \).
Here is my attempt at a solution:
\(\mathcal{C}\) is some convex subset of \(\mathbb{R}^3\).
\( \mathcal{D} = \{ \textbf{p} + \textbf{q} \mid \textbf{p}, \textbf{q} \in \partial \mathcal{C} \} \)
Prove that \( \mathcal{D} \) is convex.
Here, \( \partial \mathcal{C} \) denotes the boundary of the set \( \mathcal{C} \).
Here is my attempt at a solution: