  The dramatic dynamic slowing down associated with the glass transition is
considered by many to be related to the existence of a static length scale that
grows when temperature decreases. Defining, identifying and measuring such a
length is a subtle and non-trivial problem. Recently, two proposals, based on
very different insights regarding the relevant physics, were put forward. One
approach is based on the point-to-set correlation technique and the other on
the scale where the lowest eigenvalue of the Hessian matrix becomes sensitive
to disorder. In this Letter we present numerical evidence that the two
approaches might result in the same identical length scale. This provides
further mutual support to their relevance and, at the same time, raise
interesting theoretical questions, discussed in the conclusion, concerning the
fundamental reason for their relationship.
