<s>[INST] What is the vector $x ∈ \mathbb{R^3}$ that achieves $max||x||_1$ subject to $||x||_2 = 1$? I'm trying to answer the questions "What is the vector $x ∈ \mathbb{R^3}$
that achieves $max||x||_1$ subject to $||x||_2 = 1$?" and "What is the vector x ∈ $R^3$ that achieves $max||x||_∞$ subject to $||x||_2 = 1$?
I think the first question is asking me to find a vector with three components that will have the maximum $||x||_1$ norm value where $\sqrt{x_1^2 + x_3^2 + x_2^2} = 1$, so $x_1^2 + x_3^2 + x_2^2 = 1$. I know the The L1 norm is just the sum of the absolute values of the vector's components. After trial and error I came up with $x = [\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}]$ , but also $[-\sqrt{\frac{1}{3}}, -\sqrt{\frac{1}{3}}, -\sqrt{\frac{1}{3}}]$, and $[-\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}]$, etc. 
For my the second question, I think I need to find the vector in $\mathbb{R^3}$ that will give me the maximum value of the absolute value of the vector's components given $x_1^2 + x_3^2 + x_2^2 = 1$. I came up with $[1, 0, 0]$, $[0, 1, 0]$ , $[0, 0, 1]$, $[-1, 0, 0]$, $[0, -1, 0]$ , and $[0, 0, -1]$. 
Am I correct? Is there a more formal way to figure this out and write my solution?
 [/INST] For the solution of your first question, you should also add the follwoing points: $\left(\sqrt{\frac{1}{3}}, -\sqrt{\frac{1}{3}}, -\sqrt{\frac{1}{3}}\right), \left(\sqrt{\frac{1}{3}}, -\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}\right), \left(\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}, -\sqrt{\frac{1}{3}}\right), \left(-\sqrt{\frac{1}{3}}, -\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}\right)$ and $\left(-\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}, -\sqrt{\frac{1}{3}}\right)$.
To intuitively guess them, you need to think that both L1 and L2 are symmetric metrics. So you should consider points $(x_1, x_2, x_3)$ where $|x_1| = |x_2| = |x_3|$. The constraint of $x_1^2 + x_2^2 + x_3^2 = 1$ would give you $|x_1| = |x_2| = |x_3| = \sqrt{\frac{1}{3}}$.
For a geomteric solution, let us first consider the case in 1st octant i.e. where $x_1 \geq 0, x_2 \geq 0, $ and $x_3 \geq 0$. Thus $L_1 = |x_1| + |x_2| + |x_3|$ is equivalent to $L_1 = x_1 + x_2 + x_3$. Now geometrically, one needs to find the point where the plane $x_1 + x_2 + x_3 = constant$ touches the sphere $x_1^2 + x_2^2 + x_3^2 = 1$ in the first octant.
If you want to formally and algebraically compute the values in the 1st oactant, then you need to write the Lagrangian function $L(x_1, x_2, x_3, \lambda) = x_1 + x_2 + x_3 + \lambda(x_1^2 + x_2^2 + x_3^2 - 1)$. Now equate all the partial derivatives to 0. This will get you $\lambda = -\frac{1}{2x_1} = -\frac{1}{2x_2} = -\frac{1}{2x_3}$. This leads to $x_1 = x_2 = x_3 = \frac{1}{\sqrt{3}}$. In the other octant (where $x_1 \le 0, x_2 \geq 0, $ and $x_3 \geq 0$) the Lagrangian function will change sign: $L(x_1, x_2, x_3, \lambda) = -x_1 + x_2 + x_3 + \lambda(x_1^2 + x_2^2 + x_3^2 - 1)$. This would yield an answer of $-x_1 = x_2 = x_3 = \frac{1}{\sqrt{3}}$.
For the second problem, let us take the first case where the $L_\infty$ norm $= Max(x_1, x_2, x_3) = x_1$. Then the Lagrangian function $L(x_1, x_2, x_3, \lambda) = x_1 + \lambda(x_1^2 + x_2^2 + x_3^2 - 1)$.
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