Large signed sums of unit vectors: the first linearly dependent case

We address a problem on large signed sums of unit vectors that arose in work of Brugger, Fiedler, González Merino and Kirschbaum and was later formulated in its present form by Ambrus and Nietert. Given $d+1$ unit vectors $u_1,\ldots,u_{d+1}$ in $\mathbb R^d$, with $d\ge2$, the problem asks for the smallest possible value of \[ \max_{\varepsilon_i=\pm1} \left\|\sum_{i=1}^{d+1}\varepsilon_i u_i\right\|. \] We prove that this value is $\sqrt{d+2}$. We also determine all equality cases: up to independent sign changes and orthogonal transformations, they consist of the vertices of a centered regular simplex of positive even dimension together with an orthonormal basis of its orthogonal complement.