  The Swampland Distance Conjecture (SDC) restricts the geodesic distances that
scalars can traverse in effective field theories as they approach points at
infinite distance in moduli space. We propose that, when applied to the subset
of light fields in effective theories with scalar potentials, the SDC restricts
the amount of non-geodesicity allowed for trajectories along valleys of the
potential. This is necessary to ensure consistency of the SDC as a valid
swampland criterium at any energy scale across the RG flow. We provide a simple
description of this effect in moduli space of hyperbolic space type, and
products thereof, and obtain critical trajectories which lead to maximum
non-geodesicity compatible with the SDC. We recover and generalize these
results by expressing the SDC as a new Convex Hull constraint on trajectories,
characterizing towers by their scalar charge to mass ratio in analogy to the
Scalar Weak Gravity Conjecture. We show that recent results on the asymptotic
scalar potential of flux compatifications near infinity in moduli space
precisely realize these critical amounts of non-geodesicity. Our results
suggest that string theory flux compactifications lead to the most generic
potentials allowing for maximum non-geodesicity of the potential valleys while
respecting the SDC along them.
