# The Brunn-Minkowski Inequality: Its Refinements and Extensions Part II

Tuesday, May 5, 2015 - 2:00pm - 3:00pm

Lind 305

Arnaud Marsiglietti (University of Minnesota, Twin Cities)

My talk is related to the Brunn-Minkowski inequality, which states that for every convex sets $A,B$ in $\mathbb{R}^n$ and for every $\lambda \in [0,1]$, one has

$$ (1-\lambda)A + \lambda B^{1/n} \geq (1-\lambda) A^{1/n} + \lambda B^{1/n}, $$

where $A+B = \{a+b ; a \in A, b \in B \}$ denotes the Minkowski sum of $A$ and $B$ and where $\cdot$ denotes the volume (Lebesgue measure).

I will introduce a generalization of the Minkowski sum and prove a Brunn-Minkowski-type inequality for general measures, with respect to this new sum.

This is related to the log-Brunn-Minkowski inequality of Lutwak-Yang-Zhang

$$ (1-\lambda)A + \lambda B^{1/n} \geq (1-\lambda) A^{1/n} + \lambda B^{1/n}, $$

where $A+B = \{a+b ; a \in A, b \in B \}$ denotes the Minkowski sum of $A$ and $B$ and where $\cdot$ denotes the volume (Lebesgue measure).

I will introduce a generalization of the Minkowski sum and prove a Brunn-Minkowski-type inequality for general measures, with respect to this new sum.

This is related to the log-Brunn-Minkowski inequality of Lutwak-Yang-Zhang