Shadow systems and volumes of polar convex bodies

Mathieu Meyer, Shlomo Reisner

Research output: Contribution to journalArticlepeer-review


It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.

Original languageEnglish
Pages (from-to)129-148
Number of pages20
Issue number1
StatePublished - Jun 2006

Bibliographical note

Funding Information:
We have thus shown that b generates an extreme ray of C and that a ≡ 0. Hence, by Lemma 14, K is a triangle. h Acknowledgements. Both authors have been supported in part by the Research Network Program in Mathematics of the High Council for Scientific and Technological Cooperation between France and Israel. Additionally, Shlomo Reisner has been supported in part by NATO Collaborative Linkage Grant PST.CLG 979701.

ASJC Scopus subject areas

  • General Mathematics


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