Abstract
We define a new transform on \(\alpha \)-concave functions, which we call the \(\sharp \)-transform. Using this new transform, we prove a sharp Blaschke–Santaló inequality for \(\alpha \)-concave functions, and characterize the equality case. This extends the known functional Blaschke–Santaló inequality of Artstein-Avidan, Klartag and Milman, and strengthens a result of Bobkov. Finally, we prove that the \(\sharp \)-transform is a duality transform when restricted to its image. However, this transform is neither surjective nor injective on the entire class of \(\alpha \)-concave functions.
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Rotem, L. A sharp Blaschke–Santaló inequality for \(\alpha \)-concave functions. Geom Dedicata 172, 217–228 (2014). https://doi.org/10.1007/s10711-013-9917-3
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DOI: https://doi.org/10.1007/s10711-013-9917-3