α-concave functions and a functional extension of mixed volumes

Vitali Milman, Liran Rotem

نتاج البحث: نشر في مجلةمقالةمراجعة النظراء

ملخص

Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation ⊕ on the class of quasi-concave functions, such that every class of α-concave functions is closed under ⊕. We then define the mixed integrals, which are the polarization of the integral with respect to ⊕. We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to α-concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form.

اللغة الأصليةالإنجليزيّة
الصفحات (من إلى)1-11
عدد الصفحات11
دوريةElectronic Research Announcements in Mathematical Sciences
مستوى الصوت20
المعرِّفات الرقمية للأشياء
حالة النشرنُشِر - 2013

All Science Journal Classification (ASJC) codes

  • !!General Mathematics

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