Clathrin-mediated endocytosis (CME) involves nanoscale bending and inward budding of the plasma membrane, by which cells regulate both the distribution of membrane proteins and the entry of extracellular species(1,2). Extensive studies have shown that CME proteins actively modulate the plasma membrane curvature(1,3,4). However, the reciprocal regulation of how the plasma membrane curvature affects the activities of endocytic proteins is much less explored, despite studies suggesting that membrane curvature itself can trigger biochemical reactions(5-8). This gap in our understanding is largely due to technical challenges in precisely controlling the membrane curvature in live cells. In this work, we use patterned nanostructures to generate well-defined membrane curvatures ranging from +50 nm to -500 nm radius of curvature. We find that the positively curved membranes are CME hotspots, and that key CME proteins, clathrin and dynamin, show a strong preference towards positive membrane curvatures with a radius <200 nm. Of ten CME-related proteins we examined, all show preferences for positively curved membrane. In contrast, other membrane-associated proteins and non-CME endocytic protein caveolin1 show no such curvature preference. Therefore, nanostructured substrates constitute a novel tool for investigating curvature-dependent processes in live cells.