Definition
A convex body in Euclidean space is defined as a compact convex set with non-empty interior. If B is a centrally symmetric convex body in n-dimensional Euclidean space, the polar body Bo is another centrally symmetric body in the same space, defined as the set
The Mahler volume of B is the product of the volumes of B and Bo.
If T is a linear transformation, then ; thus applying T to B changes its volume by and changes the volume of Bo by . Thus the overall Mahler volume of B is preserved by linear transformations.
Read more about this topic: Mahler Volume
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