An unknown point $x$ is fixed on the unit sphere in $\mathbb{R}^n$ and we iterate as follows: at each step, specify a unit vector $v\in \mathbb{R}^n$ and record $\mathrm{sign}(x\cdot v).$
The goal is to design a strategy for specifying $x$ to within the smallest ball with the fewest possible steps.
With the first step you learn which of two hemispheres $x$ is in. The next step can get you to within a quarter, the next $1/8,$ and so on. With each step you are capable of cutting in half the region $x$ can be in.
A natural strategy to get a well-localized region is to first fix an orthonormal basis for $\mathbb{R}^n$ and cycle through it, each step slicing the region along the corresponding basis element's direction. After $\ell$ steps, $x$ will be known to within a simplex with area $\propto 2^{-\ell}$. But what is the size of the ball for this strategy? Is there a better strategy?