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3
votes
1answer
116 views

Are there isomeasure simplices?

Say that two polyhedra in $\mathbb{R}^3$ have isomeasures (my terminology) if they have: the same volume, the same surface area, the same sum of all edge lengths, and the same number of vertices. The ...
4
votes
2answers
387 views

The Isoperimetric problem for domains constrained to lie between two parallel planes

It is well known that for a given volume $V$, a sphere is the shape that minimizes the surface area. I am interested in the same problem under the constraint that the shape must lie between the planes ...
1
vote
1answer
122 views

Peculiar vertex-isoperimetric inequality on the discrete torus (and generalization)

Consider a discrete even torus $G=(V,E)$, i.e. the graph on $\lbrace 0,1,\dots,n-1 \rbrace^2$, $n$ even, where two vertices are connected by an edge only if they differ by 1 in only one coordinate, ...
4
votes
3answers
218 views

Perimeter/Neighborhood of a graph on grid

Hello, I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L_1$ distance is one. Now I want to claim ...
1
vote
1answer
221 views

Hypercube isoperimetric inequality for non-increasing events

It is well known that isoperimetric inequalities on a hypercube are closely related to influences, but all the theorems I'm aware of deal with monotone sets. Now suppose we have an arbitrary set $X ...
6
votes
4answers
380 views

Packing and isoperimetrics

Suppose a manufacturer bottles small units of liquid and ships them via very large trucks. If the transportation cost nothing, spherical bottles would minimize the packaging cost (isoperimetric ...