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Although your context is broader, these level sets under the Euclidean metric are known as offset polygons. Here is an image from an earlier MO question/answeran earlier MO question/answer:

 
Another useful term in this context is the *Minkowski sum*, e.g., this PDF slide presentation by Andreas Bock: [Minkowski Sums and Offsets of Polygons](http://resources.mpi-inf.mpg.de/departments/d1/teaching/ss10/Seminar_CGGC/Slides/07_Bock_MS.pdf).

Although your context is broader, these level sets under the Euclidean metric are known as offset polygons. Here is an image from an earlier MO question/answer:

 
Another useful term in this context is the *Minkowski sum*, e.g., this PDF slide presentation by Andreas Bock: [Minkowski Sums and Offsets of Polygons](http://resources.mpi-inf.mpg.de/departments/d1/teaching/ss10/Seminar_CGGC/Slides/07_Bock_MS.pdf).

Although your context is broader, these level sets under the Euclidean metric are known as offset polygons. Here is an image from an earlier MO question/answer:

 
Another useful term in this context is the *Minkowski sum*, e.g., this PDF slide presentation by Andreas Bock: [Minkowski Sums and Offsets of Polygons](http://resources.mpi-inf.mpg.de/departments/d1/teaching/ss10/Seminar_CGGC/Slides/07_Bock_MS.pdf).
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Joseph O'Rourke
Joseph O'Rourke
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Although your context is broader, these level sets under the Euclidean metric are known as offset polygons. Here is an image from an earlier MO question/answer:

 
Another useful term in this context is the *Minkowski sum*, e.g., this PDF slide presentation by Andreas Bock: [Minkowski Sums and Offsets of Polygons](http://resources.mpi-inf.mpg.de/departments/d1/teaching/ss10/Seminar_CGGC/Slides/07_Bock_MS.pdf).