  We address the general problem of estimating the probability that a real
symmetric tensor is close to rank-one tensors. Using Weyl's tube formula, we
turn this question into a differential geometric one involving the study of
metric invariants of the real Veronese variety. More precisely, we give an
explicit formula for its reach and curvature coefficients with respect to the
Bombieri-Weyl metric. These results are obtained using techniques from Random
Matrix theory and an explicit description of the second fundamental form of the
Veronese variety in terms of GOE matrices. Our findings give a complete
solution to the original problem, and in the case of rational normal curves
lead to some novel asymptotic results.
