RegionNearest[reg,p] gives a point in the region reg that is nearest the point p.
RegionNearest[reg]gives a RegionNearestFunction[…] that can be repeatedly applied to points.
Find the nearest point in the unit disk:
Find the nearest point in a MeshRegion:
With one argument you get a RegionNearestFunction:
Apply the function to lists of points to compute many nearest points:
RegionNearestFunction RegionDistance SignedRegionDistance ArgMin NArgMin