An isoperimetric inequality on the $\ell_p$ balls

Isoperimetric-type inequalities for iterated Brownian motion in R^n

We extend generalized isoperimetric-type inequalities to iterated Brownian motion over several domains in $\RR{R}^{n}$. These kinds of inequalities imply in particular that for domains of finite volume, the exit distribution and moments of the first exit time for iterated Brownian motion are maximized with the ball $D^{*}$ centered at the origin, which has the same volume as $D$

Edge-Isoperimetric Inequalities and Ball-Noise Stability: Linear Programming and Probabilistic Approaches

Let $Q_{n}^{r}$ be the graph with vertex set $\{-1,1\}^{n}$ in which two vertices are joined if their Hamming distance is at most $r$. The edge-isoperimetric problem for $Q_{n}^{r}$ is that: For every $(n,r,M)$ such that $1\le r\le n$ and $1\le M\le2^{n}$, determine the minimum edge-boundary size of a subset of vertices of $Q_{n}^{r}$ with a given size $M$. In this paper, we apply two different approaches to prove bounds for this problem. The first approach is a linear programming approach and the second is a probabilistic approach. Our bound derived by the first approach generalizes the tight…

平面上带 l_1 度量的指数测度的等周问题

Isoperimetric problem for exponential measure on the plane with l_1-metric

本文解决了平面上带 l_1 度量的指数测度等周问题，证明给定测度下单纯形或其补集具有最小边界测度，并利用沿等 l_1 距离截面的对称化方法完成证明。

Reverse isoperimetric inequalities for parallel sets

We consider the family of $r$-parallel sets in $\mathbb{R}^d$, that is sets of the form $A_r=A+rB_2^n$, where $B_2^n$ is the unit Euclidean ball and $A$ is an arbitrary Borel set. We show that the ratio between the upper surface area measure of an $r$-parallel set and its volume is upper bounded by $d/r$. Equality is achieved for $A$ being a single point. As a consequence of our main result we show that the Gaussian upper surface area measure of an $r$-parallel set is upper bounded by $18d \max(\sqrt{d},r^{-1})$. Moreover, we observe that there exists a $1$-parallel set with Gaussian surface a…

On isoperimetric sets of radially symmetric measures

We study the isoperimetric problem for the radially symmetric measures. Applying the spherical symmetrization procedure and variational arguments we reduce this problem to a one-dimensional ODE of the second order. Solving numerically this ODE we get an empirical description of isoperimetric regions of the planar radially symmetric exponential power laws. We also prove some isoperimetric inequalities for the log-convex measures. We show, in particular, that the symmetric balls of large size are isoperimetric sets for strictly log-convex and radially symmetric measures. In addition, we establis…

An isoperimetric inequality for the Wiener sausage

Let $(ξ(s))_{s\geq 0}$ be a standard Brownian motion in $d\geq 1$ dimensions and let $(D_s)_{s \geq 0}$ be a collection of open sets in $\R^d$. For each $s$, let $B_s$ be a ball centered at 0 with $\vol(B_s) = \vol(D_s)$. We show that $\E[\vol(\cup_{s \leq t}(ξ(s) + D_s))] \geq \E[\vol(\cup_{s \leq t}(ξ(s) + B_s))]$, for all $t$. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.