Abstract We prove that for any semi-norm $\|\cdot \|$ on $\mathbb{R}^n$ and symmetric convex body $K$ in $\mathbb{R}^n,$(1)$$\begin{align}& \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial \|n_x\| \right)^2, \end{align}$$and characterize the equality cases of this new inequality. The above would also follow from Log-Brunn–Minkowski conjecture if...

In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended capacities constructed authors based on a class Hamiltonian non-periodic boundary value problems recently. Then introduce billiards domains, and them prove some corresponding results those per...

We prove a sharp stability result concerning how close homothetic sets attaining near-equality in the Brunn–Minkowski inequality are to being convex. In particular, resolving conjecture of Figalli and Jerison, we show there universal constants $\\smash{C_n,d_n>0}$ such that for $\\smash{A \\subset \\mathbb{R}^n}$ positive measure, if $\\smash{|\\frac{A+A}{2}\\setminus A| \\le d_n |A|}$, then $\...