On the analogue of the concavity of entropy power in the Brunn-Minkowski theory
نویسندگان
چکیده
Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured in On the similarity of the entropy power inequality and the BrunnMinkowski inequality (IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839) the 1 n -concavity of the outer parallel volume of measurable sets as an analogue of the concavity of entropy power. We investigate this conjecture and study its relationship with geometric inequalities.
منابع مشابه
Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality
In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability (for an introduction see [9]). The analogue of entropy in the free context was introduced by the second named author in [8]. Here we show that Shannon's entropy power inequality ([6],[1]) has an analogue for the free entropy χ(X...
متن کاملForward and Reverse Entropy Power Inequalities in Convex Geometry
The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent developments on forward and reverse entropy power inequalities not just for the Shannon-Boltzmann entropy but also more generally for Rényi entropy. In the proce...
متن کاملRestricted Prékopa-leindler Inequality
We prove a functional version of the Brunn-Minkowski inequality for restricted sums obtained by Szarek and Voicu-lescu. We only consider Lebesgue-measurable subsets of R n , and for A ⊂ R n , we denote its volume by |A|. If A, B ⊂ R n , their Minkowski sum is defined by A + B = {x + y, (x, y) ∈ A × B}. The classical Brunn-Minkowski inequality provides a lower bound for its volume. In their stud...
متن کاملOn the Volume of the Minkowski Sum of Line Sets and the Entropy-Power Inequality
We derive a Brunn-Minkowski-type inequality regarding the volume of the Minkowski sum of degenerate sets, namely, line sets. Let A 1 : : : A n be one dimensional sets of unit length, and v
متن کاملDimensional behaviour of entropy and information
We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman’s reverse Brunn-M...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1302.6093 شماره
صفحات -
تاریخ انتشار 2013