Abstract: The set of quantum correlations, often denoted Q, is the collection of all possible probability distributions on measurement outcomes achievable by space-like separated parties sharing a quantum state. It is well established that this set is larger than what can be explained by classical physics. In his seminal work, Tsirelson [1] initiated the systematic study of the set of quantum correlations as a mathematical construct, which has so far mainly been conducted using algebraic and convex geometry techniques. In this work, we explore a more analytical approach to Q. We endow the set of state and measurement combinations with its canonical homogeneous Riemannian manifold structure, allowing us to consider infinitesimal perturbations of quantum strategies, and the induced response on the associated correlation. Using these new tools in the n2d scenario (n players, two measurements with d possible outcomes), we uncover a rather surprising structural result: a quantum correlation point which achieves the maximal violation of some Bell inequality cannot arise from pure qudits with projective measurements, provided it lies sufficiently close to a local deterministic point. More precisely, if a correlation point attains a strictly maximal classical value of a Bell functional g, then it is a local optimum for g in the set of correlation points realizable by pure qudit states and projective measurements. In the n22 setting, this shows, by Masanes’ theorem [2] that Q does not exhibit any extremal point around local deterministic points, meaning that the faces of Q surrounding these points are flat. Our result has practical consequences on the hardness of finding violations of Bell inequalities using gradient-based optimization of state and measurement combinations. We conjecture that the analysis can be generalized to pure quDits in the n2d scenario, with D > d, which would prove that some correlation points maximally violating a Bell inequality cannot be obtained without POVMs.