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The Swampland Distance Conjecture (SDC) restricts the geodesic distances thatscalars can traverse in effective field theories as they approach points atinfinite distance in moduli space. We propose that, when applied to the subsetof light fields in effective theories with scalar potentials, the SDC restrictsthe amount of non-geodesicity allowed for trajectories along valleys of thepotential. This is necessary to ensure consistency of the SDC as a validswampland criterium at any energy scale across the RG flow. We provide a simpledescription of this effect in moduli space of hyperbolic space type, andproducts thereof, and obtain critical trajectories which lead to maximumnon-geodesicity compatible with the SDC. We recover and generalize theseresults by expressing the SDC as a new Convex Hull constraint on trajectories,characterizing towers by their scalar charge to mass ratio in analogy to theScalar Weak Gravity Conjecture. We show that recent results on the asymptoticscalar potential of flux compatifications near infinity in moduli spaceprecisely realize these critical amounts of non-geodesicity. Our resultssuggest that string theory flux compactifications lead to the most genericpotentials allowing for maximum non-geodesicity of the potential valleys whilerespecting the SDC along them.

The Convex Hull Swampland Distance Conjecture and Bounds on Non-geodesics