Below are two known lemmas on a $d$-dimensional sphere (related to the isoperimetric inequality). I would like to know: does a similar statement like this holds for a $d$-dimensional dome also (i.e. half the sphere)

Lemma 1: Suppose $s$ is a subset of the $d$-sphere with normalised measure $\mu(s)=1/2$. The $\epsilon$-expansion of the set $s$ is then at least as large as the $\epsilon$-expansion of a half sphere, if we use the geodesic metric.

Lemma 2: An $\epsilon$-expansion of a half sphere in the geodesic metric has a normalised measure $\geq 1-\sqrt{\pi/8} \exp(-d \epsilon^2/2)$

Definition used: $\epsilon$-expansion of a set $s$ contains all points that are at most $\epsilon$ units away from $s$ according to a specified metric.

Suppose we don't have a sphere to begin with, but a $d$-dimensional dome or half a sphere instead. We can keep using the geodesic metric. Then we take a subset $s'$ of this dome with normalised measure $\mu(s')=0.5$. What is the normalised measure of the $\epsilon$-expansion of this set $s'$? Can it be lower-bounded by some function $f(d, \epsilon)$ just like lemma 2 for the sphere?

Any leads on this problem would be much appreciated! Thank you in advance!


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.