# Lower bound to $\epsilon$-expansion of a subset of a half-sphere

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!