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{{Short description|Concept in convex geometry}}
In [[convex geometry]], the '''Mahler volume''' of a [[central symmetry|centrally symmetric]] [[convex body]] is a [[dimensionless quantity]] that is associated with the body and is [[invariant (mathematics)|invariant]] under [[linear transformation]]s. It is named after German-English mathematician [[Kurt Mahler]]. It is known that the shapes with the largest possible Mahler volume are the [[ball (mathematics)|balls]] and solid [[ellipsoid]]s; 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]].

==Definition==
A convex body in [[Euclidean space]] is defined as a [[compact space|compact]] [[convex set]] with [[empty set|non-empty]] interior.  If <math>B</math> is a centrally symmetric convex body in <math>n</math>-dimensional Euclidean space, the [[polar set|polar body]] <math>B^\circ</math> is another centrally symmetric body in the same space, defined as the set
<math display=block>\left\{ x\mid x\cdot y\le 1 \text{ for all } y\in B \right\}.</math>
The Mahler volume of <math>B</math> is the product of the volumes of <math>B</math> and <math>B^\circ</math>.<ref name="tao">{{harvtxt|Tao|2007}}.</ref>

If <math>T</math> is an [[invertible]] linear transformation, then <math>(TB)^\circ = (T^{-1})^\ast B^\circ</math>. Applying <math>T</math> to <math>B</math> multiplies its volume by <math>\det T</math> and multiplies the volume of <math>B^\circ</math> by <math>\det (T^{-1})^\ast</math>.  As these determinants are [[multiplicative inverse]]s, the overall Mahler volume of <math>B</math> is preserved by linear transformations.

==Examples==
The polar body of an [[n-sphere|<math>n</math>-dimensional unit sphere]] is itself another unit sphere. Thus, its Mahler volume is just the square of its volume, 
:<math>\frac{\Gamma(3/2)^{2n}4^n}{\Gamma(\frac{n}{2}+1)^2}</math>
where <math>\Gamma</math> is the [[Gamma function]].
By affine invariance, any [[ellipsoid]] has the same Mahler volume.<ref name="tao"/>

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<ref name="tao"/>
:<math>\frac{4^n}{\Gamma(n+1)}.</math>

The Mahler volume of the sphere is larger than the Mahler volume of the hypercube by a factor of approximately <math>\left(\tfrac{\pi}{2}\right)^n</math>.<ref name="tao"/>

==Extreme shapes==
{{unsolved|mathematics|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 [[mathematical proof|proven]] by {{harvs|first=Wilhelm |last=Blaschke |authorlink=Wilhelm Blaschke|txt|year=1917}}; the full result was proven much later by {{harvs|first=Luis|last=Santaló|authorlink=Luis Santaló|txt|year=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.<ref name="tao"/>

The shapes with the minimum known Mahler volume are [[hypercube]]s, [[cross polytope]]s, and more generally the [[Hanner polytope]]s which include these two types of shapes, as well as their affine transformations. The Mahler conjecture states that the Mahler volume of these shapes is the smallest of any ''n''-dimensional symmetric convex body; it remains unsolved when <math>n\geq4</math>. As [[Terence Tao|Terry Tao]] writes:<ref name="tao"/>
{{cquote|The main reason why this [[conjecture]] is so difficult is that unlike the upper [[upper and lower bounds|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.}}

{{harvtxt|Bourgain|Milman|1987}} proved that the Mahler volume is bounded below by <math>c^n</math> times the volume of a sphere for some absolute constant <math>c > 0</math>, matching the scaling behavior of the hypercube volume but with a smaller constant. {{harvtxt|Kuperberg|2008}} proved that, more concretely, one can take <math>c=\tfrac{1}{2}</math> 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<ref>{{Cite journal|last1=Mahler|first1=Kurt|author-link=Kurt Mahler|title=Ein Minimalproblem für konvexe Polygone|journal=Mathematica (Zutphen) B|pages=118–127|year=1939}}</ref> and the 3-dimensional case by Iriyeh and Shibata.<ref>{{Cite journal|last1=Iriyeh|first1=Hiroshi|last2=Shibata|first2=Masataka|title=Symmetric Mahler's conjecture for the volume product in the 3-dimensional case|journal=[[Duke Mathematical Journal]]|volume=169|issue=6|pages=1077–1134|arxiv=1706.01749|doi=10.1215/00127094-2019-0072|year=2020|mr=4085078}}</ref>
* It is known that each of the [[Hanner polytope]]s is a strict local minimizer for the Mahler volume in the class of origin-symmetric convex bodies endowed with the [[Banach–Mazur distance]]. This was first proven by Nazarov, Petrov, Ryabogin, and Zvavitch for the unit cube,<ref>{{harvtxt|Nazarov|Petrov|Ryabogin|Zvavitch|2010}}</ref> and later generalized to all Hanner polytopes by Jaegil Kim.<ref>{{cite journal
| last1=Kim | first1=Jaegil
| date=2014
| title=Minimal volume product near Hanner polytopes
| journal=Journal of Functional Analysis
| volume=266
| issue=4
| pages=2360–2402
| doi=10.1016/j.jfa.2013.08.008 | doi-access=free| arxiv=1212.2544
}}</ref>
* The Mahler conjecture holds for [[zonotope]]s.<ref>{{cite journal
| last1=Reisner | first1=Shlomo
| date=1986
| title=Zonoids with minimal volume-product
| journal=Mathematische Zeitschrift
| volume=192
| issue=3
| pages=339–346
| doi=10.1007/BF01164009}}</ref>
* The Mahler conjecture holds in the class of [[unconditional bodies]], that is, convex bodies invariant under reflection on each coordinate hyperplane {''x''<sub>''i''</sub> = 0}. This was first proven by Saint-Raymond in 1980.<ref>{{cite journal
| last1=Saint-Raymond | first1=J.
| date=1980
| title=Sur le volume des corps convexes symétriques
| journal=Séminaire d'initiation à l'analyse
| volume=81}}</ref> Later, a much shorter proof was found by Meyer.<ref>{{cite journal
| last1=Meyer | first1=Mathieu
| date=1986
| url=https://link.springer.com/article/10.1007/BF02765029
| title=Une caractérisation volumique de certains espaces normés de dimension finie
| journal=[[Israel Journal of Mathematics]]
| volume=55
| issue=3
| pages=317–326
| doi=10.1007/BF02765029}}</ref> This was further generalized to convex bodies with [[symmetry (geometry)|symmetry groups]] that are more general [[reflection group]]s. The minimizers are then not necessarily Hanner polytopes, but were found to be regular polytopes corresponding to the reflection groups.<ref>{{cite journal
| last1=Barthe | first1=F.
| last2=Fradelizi | first2=M.
| title=The volume product of convex bodies with many symmetries
| journal=American Journal of Mathematics
| volume=135
| issue=2
| date=April 2013
| pages=311–347
| doi=10.1353/ajm.2013.0018
| jstor=23525797}}</ref>
* Reisner et al. (2010) showed that a minimizer of the Mahler volume must have [[Gaussian curvature]] equal to zero almost everywhere on its boundary, suggesting strongly that a minimal body is a polytope.<ref>{{cite journal
| title=Mahler's Conjecture and Curvature
| last1=Reisner | first1=Shlomo
| last2=Schütt | first2=Carsten
| last3=Werner | first3=Elisabeth M.
| journal=International Mathematics Research Notices
| volume=2012
| issue=1
| date=2012
| pages=1–16
| doi=10.1093/imrn/rnr003
| arxiv=1009.3583}}</ref>

==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.{{sfnp|Kuperberg|2008}}

==Notes==
{{reflist}}

==References==
*{{cite journal
| last1= Blaschke | first1= Wilhelm | author-link1= Wilhelm Blaschke
| date= 1917
| title= Uber affine Geometrie VII: Neue Extremeingenschaften von Ellipse und Ellipsoid
| journal= Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl.
| language= German| location= Leipzig | volume= 69| pages= 412–420}}
*{{cite journal
 | last1 = Bourgain | first1 = Jean |author1-link=Jean Bourgain
 | last2 = Milman | first2 = Vitali D.
 | doi = 10.1007/BF01388911
 | mr = 880954
 | issue = 2
 | journal = [[Inventiones Mathematicae]]
 | pages = 319–340
 | title = New volume ratio properties for convex symmetric bodies in <math>\R^n</math>
 | volume = 88
 | year = 1987| bibcode = 1987InMat..88..319B }}
*{{cite journal
 | last = Kuperberg | first = Greg | author-link = Greg Kuperberg
 | arxiv = math/0610904
 | doi = 10.1007/s00039-008-0669-4
 | issue = 3
 | journal = [[Geometric and Functional Analysis]]
 | mr = 2438998
 | pages = 870–892
 | title = From the Mahler conjecture to Gauss linking integrals
 | volume = 18
 | year = 2008}}
*{{Cite journal|last1=Nazarov|first1=Fedor|last2=Petrov|first2=Fedor|
first3=Dmitry|last3= Ryabogin|first4= Artem|last4= Zvavitch|title=A remark on the Mahler conjecture: local minimality of the unit cube|journal=[[Duke Mathematical Journal]]|volume=154|issue=3|pages=419–430|arxiv=0905.0867|doi=10.1215/00127094-2010-042|year=2010|mr=2730574}}
*{{cite journal
 | last = Santaló | first = Luis A.|author-link = Luis Santaló 
 | mr = 0039293
 | journal = [[Portugaliae Mathematica]]
 | pages = 155–161
 | language = es
 | title = An affine invariant for convex bodies of <math>n</math>-dimensional space
 | volume = 8
 | year = 1949
}}
*{{cite web|url=http://terrytao.wordpress.com/2007/03/08/open-problem-the-mahler-conjecture-on-convex-bodies/|first=Terence|last=Tao|author-link=Terence Tao|title=Open question: the Mahler conjecture on convex bodies|date=March 8, 2007}} Revised and reprinted in {{cite book|first=Terence|last=Tao|author-link=Terence Tao|contribution=3.8 Mahler's conjecture for convex bodies|pages=216–219|title=Structure and Randomness: Pages from Year One of a Mathematical Blog|publisher=[[American Mathematical Society]]|year=2009|isbn=978-0-8218-4695-7}}

{{Convex analysis and variational analysis}}
{{Functional analysis}}

[[Category:Convex geometry]]
[[Category:Geometric inequalities]]
[[Category:Volume]]