Mahler volume
In
Definition
A convex body in Euclidean space is defined as a compact convex set with non-empty interior. If is a centrally symmetric convex body in -dimensional Euclidean space, the polar body is another centrally symmetric body in the same space, defined as the set
If is an
Examples
The polar body of an -dimensional unit sphere is itself another unit sphere. Thus, its Mahler volume is just the square of its volume,
where is the Gamma function. By affine invariance, any ellipsoid has the same Mahler volume.[1]
The polar body of a
The Mahler volume of the sphere is larger than the Mahler volume of the hypercube by a factor of approximately .[1]
Extreme shapes
Is the Mahler volume of a centrally symmetric convex body always at least that of the hypercube of the same dimension?
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
The shapes with the minimum known Mahler volume are
The main reason why this conjecture is so difficult is that unlike the upper bound, in which there is essentially only one extremiser up to affine transformations (namely the ball), there are many distinct extremisers for the lower bound - not only the cube and the octahedron, but also products of cubes and octahedra, polar bodies of products of cubes and octahedra, products of polar bodies of… well, you get the idea. It is really difficult to conceive of any sort of flow or optimisation procedure which would converge to exactly these bodies and no others; a radically different type of argument might be needed.
Bourgain & Milman (1987) proved that the Mahler volume is bounded below by times the volume of a sphere for some absolute constant , matching the scaling behavior of the hypercube volume but with a smaller constant. Kuperberg (2008) proved that, more concretely, one can take in this bound. A result of this type is known as a reverse Santaló inequality.
Partial results
- The 2-dimensional case of the Mahler conjecture has been solved by Mahler (1939) and the 3-dimensional case by Iriyeh & Shibata (2020).
- Banach–Mazur distance.
For asymmetric bodies
The Mahler volume can be defined in the same way, as the product of the volume and the polar volume, for convex bodies whose interior contains the origin regardless of symmetry. Mahler conjectured that, for this generalization, the minimum volume is obtained by a simplex, with its centroid at the origin. As with the symmetric Mahler conjecture, reverse Santaló inequalities are known showing that the minimum volume is at least within an exponential factor of the simplex.[2]
Notes
- ^ a b c d e f Tao (2007).
- ^ Kuperberg (2008).
References
- Blaschke, Wilhelm (1917). "Uber affine Geometrie VII: Neue Extremeingenschaften von Ellipse und Ellipsoid". Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. (in German). 69. Leipzig: 412–420.
- Bourgain, Jean; Milman, Vitali D. (1987). "New volume ratio properties for convex symmetric bodies in ". MR 0880954.
- Iriyeh, Hiroshi; Shibata, Masataka (2020). "Symmetric Mahler's conjecture for the volume product in the 3-dimensional case". MR 4085078.
- MR 2438998.
- Mahler, Kurt (1939). "Ein Minimalproblem für konvexe Polygone". Mathematica (Zutphen) B: 118–127.
- Nazarov, Fedor; Petrov, Fedor; Ryabogin, Dmitry; Zvavitch, Artem (2010). "A remark on the Mahler conjecture: local minimality of the unit cube". MR 2730574.
- Santaló, Luis A. (1949). "An affine invariant for convex bodies of -dimensional space". MR 0039293.
- ISBN 978-0-8218-4695-7.