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You might look at the CGAL Manual, Section 24.3 "Offsetting a Polygon," which describes how to compute the offset curve for a simple polygon via convolutions. The left image below shows the basic idea (and also serves to illustrate Bill Thurston's point about drawing equal-radii circles), and the right image illustrates a challenging example. [Both images from the CGAL manual.] Offset curves are used in many practical contexts (e.g., numerically controlled (CNC) milling machines), and so have been studied intensively. Despite that study, there is no simple way known to compute the offset.
Offset Polygons http://cs.smith.edu/%7Eorourke/MathOverflow/OffsetPolygons.jpg
To further supplement Bill's answer, Adobe Illustrator has an "offset-curve" option: Select the curve, select "Object | Path > Offset Path ...", enter the radius, and you get something like this:
           Offsets http://cs.smith.edu/%7Eorourke/MathOverflow/IllustratorOffset.jpg