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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!

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