  We study streaming algorithms for the $\ell_p$ subspace approximation
problem. Given points $a_1, \ldots, a_n$ as an insertion-only stream and a rank
parameter $k$, the $\ell_p$ subspace approximation problem is to find a
$k$-dimensional subspace $V$ such that $(\sum_{i=1}^n d(a_i, V)^p)^{1/p}$ is
minimized, where $d(a, V)$ denotes the Euclidean distance between $a$ and $V$
defined as $\min_{v \in V}\|{a - v}\|_{\infty}$. When $p = \infty$, we need to
find a subspace $V$ that minimizes $\max_i d(a_i, V)$. For $\ell_{\infty}$
subspace approximation, we give a deterministic strong coreset construction
algorithm and show that it can be used to compute a $\text{poly}(k, \log n)$
approximate solution. We show that the distortion obtained by our coreset is
nearly tight for any sublinear space algorithm. For $\ell_p$ subspace
approximation, we show that suitably scaling the points and then using our
$\ell_{\infty}$ coreset construction, we can compute a $\text{poly}(k, \log n)$
approximation. Our algorithms are easy to implement and run very fast on large
datasets. We also use our strong coreset construction to improve the results in
a recent work of Woodruff and Yasuda (FOCS 2022) which gives streaming
algorithms for high-dimensional geometric problems such as width estimation,
convex hull estimation, and volume estimation.
