Summary: A Hilbert point in \(H^p(\mathbb{T}^d)\), for \(d\geq 1\) and \(1\leq p \leq \infty \), is a nontrivial function \(\varphi\) in \(H^p(\mathbb{T}^d)\) such that \(\| \varphi \|_{H^p(\mathbb{T}^d)} \leq \|\varphi + f\|_{H^p(\mathbb{T}^d)}\) whenever \(f\) is in \(H^p(\mathbb{T}^d)\) and orthogonal to \(\varphi\) in the usual \(L^2\) sense. When \(p\neq 2\), \(\varphi\) is a Hilbert point in \(H^p(\mathbb{T})\) if and only if \(\varphi\) is a nonzero multiple of an inner function. An inner function on \(\mathbb{T}^d\) is a Hilbert point in any of the spaces \(H^p(\mathbb{T}^d)\), but there are other Hilbert points as well when \(d\geq 2\). The case of 1-homogeneous polynomials is studied in depth and, as a byproduct, a new proof is given for the sharp Khinchin inequality for Steinhaus variables in the range \(2<p<\infty \). Briefly, the dynamics of a certain nonlinear projection operator is treated. This operator characterizes Hilbert points as its fixed points. An example is exhibited of a function \(\varphi\) that is a Hilbert point in \(H^p(\mathbb{T}^3)\) for \(p=2, 4\), but not for any other \(p\); this is verified rigorously for \(p>4\) but only numerically for \(1\leq p<4\).