The following answers questions 1 to 4.

The misfit and the oriented misfit can be arbitrarily large. Fix $A$, and let $B$ be a very long (and thin so that it has volume $1$) cylinder. Then the volume of $A+B$ tends to infinity with the length of the cylinder (because $A+B$ contains many copies of $A$).

The following answers questions 1 to 4.

The misfit and the oriented misfit can be arbitrarily large. Let $A$ be a ball, and let $B$ be a very long (and thin so that it has volume $1$) cylinder. We have $|G(A)+B| = |A+B|$. The volume of $A+B$ tends to infinity with the length of the cylinder (because $A+B$ contains many copies of $A$).

The misfit and the oriented misfit can be arbitrarily large. Fix $A$, and let $B$ be a very long (and thin so that it has volume $1$) cylinder. Then the volume of $A+B$ tends to infinity with the length of the cylinder (because $A+B$ contains many copies of $A$.

The following answers questions 1 to 4.

The misfit and the oriented misfit can be arbitrarily large. Fix $A$, and let $B$ be a very long (and thin so that it has volume $1$) cylinder. Then the volume of $A+B$ tends to infinity with the length of the cylinder (because $A+B$ contains many copies of $A$).