17 сентября (в четверг) в 17:00 состоится научный семинар кафедры высшей математики МФТИ в дистанционном режиме.
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Докладчик: Г.М. Иванов
Тема доклада: "The John ellipsoid for log-concave functions"
Аннотация:
We extend the notion of the John ellipsoid (the largest volume ellipsoid contained within a convex body) to the setting of log-concave functions. For each s > 0, we define a class of log-concave functions derived from ellipsoids. For any log-concave function f, and any fixed s > 0, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or equal to f . We show that it exists, and is unique, and call it the John s-function of f. We prove a necessary and sufficient condition for the unit ball to be the John s-ellipsoid of an integrable log-concave function. As an application, we prove a quantitative Helly-type result about the integral of the pointwise minimum of a family of log-concave functions (to the best of our knowledge, it is the first result of this type). We will discuss the properties of John s-functions as s tends to zero and to infinity.