Geodesic distance and Monge-Ampère measures on contact sets

Abstract: We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the $K$-energy in this general setting. We then study Monge-Amp`ere measures on contact sets, generalizing a recent result by the first author and Trapani.