I would like to minimize $$\sum^N_i \sqrt[2^i]{x_i} = x_0 + \sqrt{x_1} + \sqrt[4]{x_2} + \cdots$$ subject to the constraints $$1 = \prod^N_i x_i = x_0x_1x_2\cdots $$ and $$x_i \in (0, 1]$$
Is this even a well defined problem? I am a bit out of my mathematical depth here.
I know if I didn't have the roots, then I'm asking how to minimize the sum of side lengths of a hypercube with constant volume. That problem is solved by setting all sides equal to one another.
For N=2:
$$ x_1 = \frac{1}{x_0} \\ x_0 + x_0^{-1/2} $$
Take a derivative, set to zero (knowing $x_0$ cannot be zero):
$$ 0 = 1 + 1/2 * x_0^{-3/2} \\ {(-2)}^{-2/3} = x_0 \\ 1/\sqrt[3]{4} = x_0 = 0.629\cdots $$
But now $x_1$ is outside the domain!
Another way I thought to approach this was starting with all set to 1. How would I perturb this to improve my minimum? If I halve $x_0$, I must double $x_1$. The sum is now:
$$ 1/2 + \sqrt{2} = \frac{1 + 2\sqrt{2}}{2} $$
Which is indeed slightly less than the original sum of two. I suppose I could repeat this ad infinitum, and bring $x_0$ as close to zero as possible?
Any pointers on how to tackle this problem are much appreciated!
Hint
$$x_0+x_1^{\frac 12}+x_2^{\frac 14}+x_3^{\frac 18}+...+x_N^{\frac{1}{2^N}}=$$$$x_0+\frac 12x_1^{\frac 12}+\frac 12x_1^{\frac 12}+\frac 14x_2^{\frac 14}+\frac 14x_2^{\frac 14}+\frac 14x_2^{\frac 14}+\frac 14x_2^{\frac 14}+..$$
Now by $AM-GM$:
$x_0+\frac 12x_1^{\frac 12}+\frac 12x_1^{\frac 12}+\frac 14x_2^{\frac 14}+\frac 14x_2^{\frac 14}+\frac 14x_2^{\frac 14}+\frac 14x_2^{\frac 14}+..\geq (1+2+2^2+...+2^N)\sqrt[1+2+2^2+...+2^N]{x_0x_1x_2...x_n \prod_{i=1}^{n} \frac{1}{2^{2^i}}}$$