Equivalence of norms => Maximization of Vectors

Question

my intuition says I'm right, but I couldn't find or get a proof for it:

Suppose I have a vector function $\vec{x}(t) \in \mathbb{R}^n$. I am interested in finding $\mathrm{argmax}_t ||\vec{x}(t)||_2$. For some reasons, this optimization problem is hard to solve, what I can solve easily, however, is $\mathrm{argmax}_t ||\vec{x}(t)||_1$.

I know that 1-norm and 2-norm are equivalent in $\mathbb{R}^n$. Does this also include that the solution of these two problems are equivalent?

Answer 1

No, the two problems are not equivalent.

Take, for example, $\vec x(t)=(\cos t, \sin t)$ to see why:

• $\max \|\vec x(t)\|_1$ is attained at $t=\frac{\pi}{4} + k\frac{\pi}{2}$ for $k\in\mathbb Z$
• $\max \|\vec x(t)\|_2$ is attained at all $t\in\mathbb R$

Clearly, the solutions to these two problems are not the same.