In receptor-mediated endocytosis, receptor molecules initially project to the outside of the cell. Where do they end up after endocytosis?

On the inside surface of the vesicle.

Explanation:

Endocytosis may be defined as the cellular process of getting material inside the cells. The materail buds inside the cell, pinched off and taken by the cell. Foreign material and pathogens can be internalized in the cell.

The receptor mediated endocytosis requires the special protein known as clathrin for the internalization of the product. The vesicles forms in this process that contains the material and this material is specific mediated by its receptors. At the end of the process, the receptor move inside the cell and can be recycled for further process.

Thus, the answer is on the inside surface of the vesicle.