The Blaschke-Santalo Inequality is a result of convex geometry concerning the Mahler volume, a dimensionless quantity associated with a centrally symmetric convex body that remains unchanged under linear transformations. The inequality establishes that, among all centrally symmetric convex bodies, the ball and the solid ellipsoids attain the largest possible Mahler volume, making them the extremal shapes for this quantity.
Facts
StatementAmong all centrally symmetric convex bodies, the ball and solid ellipsoids attain the largest possible Mahler volume. 2 Classification
Statement Form Connections
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In Branch
Source Wikipedia: Mahler volume
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Source Blaschke-Santalo inequality, Wikipedia
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1. Wikipedia: Mahler volume
WikipediaLead section, statement-form reference
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.
In 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
View the Source 2. Blaschke-Santalo inequality, Wikipedia
Lead section
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.
Extreme shapes
The three-dimensional case of this result was proven by Wilhelm Blaschke (1917); 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.
Proved By: Wilhelm Blaschke, Lead
The three-dimensional case of this result was proven by Wilhelm Blaschke (1917); 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
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