Introduces the L^p-Legendre transform and proves a functional L^p-Santaló inequality: after translation by the L^p-Santaló point, the L^p-Mahler integral of any convex function is bounded by that of the Gaussian |x|^2/2.
Duality and Heat flow
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abstract
We reveal the relation between the Legendre transform of convex functions and heat flow evolution, and how it applies to the functional Blaschke-Santalo inequality. We also describe local maximizers in this inequality.
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math.FA 1years
2024 1verdicts
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$L^p$-Legendre Transforms and Mahler integrals: Asymptotics and the Fokker--Planck heat flow
Introduces the L^p-Legendre transform and proves a functional L^p-Santaló inequality: after translation by the L^p-Santaló point, the L^p-Mahler integral of any convex function is bounded by that of the Gaussian |x|^2/2.