You know, I cant find a tool in Illustrator to "push" (bulge) a closed path (shape). The closest is putting another copy of the object behind and putting a large stroke on it, which distorts when you get to larger inputs.
If your not familiar with push, I mean to expand the shape so that it doesn't scale, more its shape changes to be equal distance around all parts, if you had another copy of the original object in front of it.
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I'll get back to you if you haven't found it yourself
In that same menu you can also outline your strokes