Mahler volume

In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the balls and solid ellipsoids; this is now known as the Blaschke–Santaló inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube. In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the balls and solid ellipsoids; this is now known as the Blaschke–Santaló inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube. 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 an invertible linear transformation, then ( T B ) ∘ = ( T − 1 ) ∗ B ∘ {displaystyle (TB)^{circ }=(T^{-1})^{ast }B^{circ }} ; thus applying T to B changes its volume by det T {displaystyle det T} and changes the volume of Bo by det ( T − 1 ) ∗ {displaystyle det(T^{-1})^{ast }} . Thus the overall Mahler volume of B is preserved by linear transformations. The polar body of an n-dimensional unit sphere is itself another unit sphere. Thus, its Mahler volume is just the square of its volume, Here Γ represents the Gamma function.By affine invariance, any ellipsoid has the same Mahler volume. The polar body of a polyhedron or polytope is its dual polyhedron or dual polytope. In particular, the polar body of a cube or hypercube is an octahedron or cross polytope. Its Mahler volume can be calculated as The Mahler volume of the sphere is larger than the Mahler volume of the hypercube by a factor of approximately ( π 2 ) n {displaystyle left({ frac {pi }{2}} ight)^{n}} . The Blaschke–Santaló inequality states that the shapes with maximum Mahler volume are the spheres and ellipsoids. The three-dimensional case of this result was proven by Wilhelm Blaschke; the full result was proven much later by Luis Santaló (1949) using a technique known as Steiner symmetrization by which any centrally symmetric convex body can be replaced with a more sphere-like body without decreasing its Mahler volume.