The Mahler volume is a dimensionless quantity associated with a centrally symmetric convex body in convex geometry, invariant under linear transformations and named after Kurt Mahler. It is known that the largest possible Mahler volume is attained by balls and solid ellipsoids, a result called the Blaschke-Santalo inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is instead attained by a hypercube. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe still-unsolved Mahler conjecture states that the minimum possible Mahler volume of a centrally symmetric convex body is attained by a hypercube. 1 Proposed Year Progress Toward ResolutionThe 2-dimensional case has been solved by Mahler and the 3-dimensional case by Iriyeh and Shibata; the general conjecture remains unsolved for dimension 4 and above. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Wikipedia: Mahler volume
Posed By
Source Wikipedia: Mahler volume
Sources
1. Mahler Conjecture (Wikipedia)
Wikimedia FoundationLead section
The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube.
Lead section, conjecture statement sentence
The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube.
Partial results section
The 2-dimensional case of the Mahler conjecture has been solved by Mahler and the 3-dimensional case by Iriyeh and Shibata.
Extreme shapes section
remains unsolved when n ≥ 4
View the Source 2. A connection between the Mahler conjecture and floating bodies (arXiv preprint)
Section 1.1, Mahler's ConjectureQuote, Section 1.1, Mahler's Conjecture
Ever since Mahler's conjecture was first stated in 1939
Wikipedia: Mahler volume
WikipediaIn Branch: Discrete Geometry, Lead sentence
In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the
Posed By: Kurt Mahler, Lead paragraph
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
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