A sharp Blaschke–Santaló inequality for α-concave functions

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We define a new transform on α-concave functions, which we call the ♯-transform. Using this new transform, we prove a sharp Blaschke–Santaló inequality for α-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 ♯-transform is a duality transform when restricted to its image. However, this transform is neither surjective nor injective on the entire class of α-concave functions.

Original languageEnglish
Pages (from-to)217-228
Number of pages12
JournalGeometriae Dedicata
Issue number1
StatePublished - 1 Oct 2014
Externally publishedYes


  • Blaschke–Santaló inequality
  • Convexity
  • Log-concavity
  • α-Concavity

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


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