TY - JOUR
T1 - Improved bounds for Hermite-Hadamard inequalities in higher dimensions
AU - Brandolini, Barbara
AU - Brandolini, Barbara
AU - Steinerberger, Stefan
AU - Langford, Jeffrey J.
AU - Larson, Simon
AU - Beck, Thomas
AU - Steinerberger, Stefan
AU - Henrot, Antoine
AU - Burdzy, Krzysztof
AU - Smits, Robert
PY - 2019
Y1 - 2019
N2 - Let $Omega subset mathbb{R}^n$ be a convex domain and let $f:Omega ightarrow mathbb{R}$ be a positive, subharmonic function (i.e. $Delta f geq 0$). Then$$ rac{1}{|Omega|} int_{Omega}{f dx} leq rac{c_n}{ |partial Omega| } int_{partial Omega}{ f dsigma},$$where $c_n leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We alsoshow that the optimal constant satisfies $c_n geq n-1$. As a byproduct,we establish a sharp geometric inequality for two convex domains where one contains the other $ Omega_2 subset Omega_1 subset mathbb{R}^n$: $$ rac{|partial Omega_1|}{|Omega_1|} rac{| Omega_2|}{|partial Omega_2|} leq n.$$
AB - Let $Omega subset mathbb{R}^n$ be a convex domain and let $f:Omega ightarrow mathbb{R}$ be a positive, subharmonic function (i.e. $Delta f geq 0$). Then$$ rac{1}{|Omega|} int_{Omega}{f dx} leq rac{c_n}{ |partial Omega| } int_{partial Omega}{ f dsigma},$$where $c_n leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We alsoshow that the optimal constant satisfies $c_n geq n-1$. As a byproduct,we establish a sharp geometric inequality for two convex domains where one contains the other $ Omega_2 subset Omega_1 subset mathbb{R}^n$: $$ rac{|partial Omega_1|}{|Omega_1|} rac{| Omega_2|}{|partial Omega_2|} leq n.$$
UR - http://hdl.handle.net/10447/493949
M3 - Article
VL - 31
SP - 801
EP - 816
JO - THE JOURNAL OF GEOMETRIC ANALYSIS
JF - THE JOURNAL OF GEOMETRIC ANALYSIS
SN - 1050-6926
ER -