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Polar Body

The polar body of a convex body K subset R^n containing the origin in its interior is

K degrees={y in R^n:<x,y><=1 for every x in K},

where <x,y> is the dot product. It is again a convex body, and (K^ degrees) degrees=K. For example, the polar body of the n-dimensional cube [-1,1]^n is the cross polytope {y:sum_(i=1)^(n)|y_i|<=1}.

The Mahler volume product conjecture concerns the product of the volumes of a convex body and an appropriately centered polar body.

See also

Convex Body, Cross Polytope, Mahler Volume Product Conjecture, Metric Entropy Duality Conjecture, Santalo Point, Translative Covering Number, Vector Space Polar

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References

Fradelizi, M.; Meyer, M.; and Zvavitch, A. "Volume Product." In Harmonic Analysis and Convexity (Ed. A. Koldobsky and A. Volberg). Berlin, Germany: De Gruyter, pp. 163-222, 2023. https://doi.org/10.1515/9783110775389-005.

Cite this as:

Weisstein, Eric W. "Polar Body." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolarBody.html

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