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A Remarkable Measure Preserving Diffeomorphism between two Convex Bodies in ℝn

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Abstract

We prove that for any two convex open bounded bodies K and T there exists a diffeomorphism f : K → T preserving volume ratio (i.e. with constant determinant of the Jacobian) and such that the Minkowski sum K + T { x + f (x) | x ∈ K }. As an application of this method, we prove some of the Alexandov–Fenchel inequalities.

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Alesker, S., Dar, S. & Milman, V. A Remarkable Measure Preserving Diffeomorphism between two Convex Bodies in ℝn . Geometriae Dedicata 74, 201–212 (1999). https://doi.org/10.1023/A:1005087216335

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