  Given a finite set $X \subset \mathbb{R}^d$ and a binary linear classifier
$c: \mathbb{R}^d \to \{0,1\}$, how many queries of the form $c(x)$ are required
to learn the label of every point in $X$? Known as \textit{point location},
this problem has inspired over 35 years of research in the pursuit of an
optimal algorithm. Building on the prior work of Kane, Lovett, and Moran (ICALP
2018), we provide the first nearly optimal solution, a randomized linear
decision tree of depth $\tilde{O}(d\log(|X|))$, improving on the previous best
of $\tilde{O}(d^2\log(|X|))$ from Ezra and Sharir (Discrete and Computational
Geometry, 2019). As a corollary, we also provide the first nearly optimal
algorithm for actively learning halfspaces in the membership query model. En
route to these results, we prove a novel characterization of Barthe's Theorem
(Inventiones Mathematicae, 1998) of independent interest. In particular, we
show that $X$ may be transformed into approximate isotropic position if and
only if there exists no $k$-dimensional subspace with more than a
$k/d$-fraction of $X$, and provide a similar characterization for exact
isotropic position.
