Can I round the corners of a shape that has curves in Illustrator?

I have a simple shape in Illustrator that has two curved edges, two straight edges, and four sharp corners:

I want to round off the sharp corners a bit like so (kinda sorta approximated by the Feather effect).

Effect -> Stylize -> Round Corners doesn't work because it gets rid of the curved edges and makes straight lines instead.

Is there a way to have Illustrator round the corners of a shape with curves?

Answer

There's no inherent method within Illustrator to do this. At least not well. You'll have to manually add anchors and create curves.

You could draw circles at the intersections and then use the Shape Builder Tool to remove the outer points..... Option/Alt-clicking those points at the corners will remove them using the Shape Builder Tool after the circles are in place. The size of the circles would determine how round the corner is.

Better solutions may be to use the Round Any Corner script, which is free.

Or look into getting the VectorScribe plug in from Astute Graphics. It's not free, but it is absolutely worth the cost.