关于词重叠的等周不等式

An isoperimetric inequality for antipodal subsets of the discrete cube

A family of subsets of $\{1,2,\ldots,n\}$ is said to be {\em antipodal} if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of $\{1,2,\ldots,n\}$. Our inequality implies that for any $k \in \mathbb{N}$, among all such families of size $2^k$, a family consisting of the union of a $(k-1)$-dimensional subcube and its antipode has the smallest possible edge boundary.

Edge Isoperimetric Inequalities for Powers of the Hypercube

For positive integers $n$ and $r$, we let $Q_n^r$ denote the $r$th power of the $n$-dimensional discrete hypercube graph, i.e., the graph with vertex-set $\{0,1\}^n$, where two 0-1 vectors are joined if they are Hamming distance at most $r$ apart. We study edge isoperimetric inequalities for this graph. Harper, Bernstein, Lindsey and Hart proved a best-possible edge isoperimetric inequality for this graph in the case $r=1$. For each $r \geq 2$, we obtain an edge isoperimetric inequality for $Q_n^r$; our inequality is tight up to a constant factor depending only upon $r$. Our techniques also yi…

A Resistance Bound via an Isoperimetric Inequality

An isoperimetric upper bound on the resistance is given. As a corollary we resolve two problems, regarding mean commute time on finite graphs and resistance on percolation clusters. Further conjectures are presented.

A tropical isoperimetric inequality

We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a higher-dimensional generalization leads to an interesting class of ordinary convex polytopes, characterizing the equality case in the isoperimetric inequality. This study is motivated by open complexity questions concerning linear optimization and its tropical analogs.

An isoperimetric inequality for planar triangulations

We prove a discrete analogue to a classical isoperimetric theorem of Weil for surfaces with non-positive curvature. It is shown that hexagons in the triangular lattice have maximal volume among all sets of a given boundary in any triangulation with minimal degree 6.

$\ell_{\infty}$ 立方体与环面上的等周不等式

Isoperimetric Inequalities in the $\ell_{\infty}$ cube and torus

本文证明了 $\ell_{\infty}$ 度量下立方体与环面的精确等周不等式，并作为推论恢复了 Bollobás-Leader 网格边等周不等式。

非局部双轴离散周长的等周不等式

Isoperimetric inequality for nonlocal bi-axial discrete perimeter

本文首次提出非局部双轴离散周长的等周不等式问题，并将其与长程双轴伊辛模型的不稳定行为研究联系起来。