50 28. SPECIAL ANCIENT SOLUTIONSConsider the following notion of the area of the boundary of a set.DEFINITION 28.33. The geometric perimeter of a measurable set E in (Mn, g)
is
(28.33) P GEo(E) = inf liminf Area(8Ei),
{Ei} i--+oo
where the infimum is taken over all sequences { Ei} of open subsets with C^00 bound-ary which volume converges to E. If n is an open set, define the relative geo-
metric perimeter of E with respect to n to be
(28.34) P GEO(E, Sl) = inf lim inf Area(8Ei n D).
{E;} i--+oo
The functions EH p GEo(E) and EH p GEo(E, n) are lower semicontinuous
with respect to volume convergence.Given x E M, we say that a unit vector v E TxM is normal to a measurable
subset E c M if
andlim Vol(E n { expx V : (V, v) < 0, IVI < c:}) = Wn
~o ~ 2lim Vol(E n { expx V : (V, v) ~ 0, IVI < c:}) =
0
e:--+0 c:n '
where Wn is the volume of the unit Euclidean n-ball; note that v is unique if it
exists. The reduced boundary of Eis the set 8* E of all points x in M such that
there exists a normal unit vector in TxM. Clearly, 8E* C 8E.
Let H k denote the k-dimensional Hausdorff measure (see, e.g., p. 100 of [39]).
We have the following result (see p.112 of [39]).LEMMA 28.34. If E is a measurable set, thenPGEo(E) = Hn-^1 (8* E):::; 1ln-^1 (8E).
Similarly, if n is an open set, then
PcEo(E, D) = Hn-^1 (8* En D):::; 1ln-^1 (8E n D).
REMARK 28.35. There exists open sets E with compact closure, with PcEo(E)< oo, and with 1ln(8E) > 0, so that 1ln-^1 (8E) = oo.
Let B ( r) denote a ball of radius r in JR^2. The following is the n = 2 and K = 0
special case of Theorem 18.l.3 in Burago and Zalgaller's book [39]. Since n = 2,
we use Area= 1l^2 to denote Vol = Hn.
THEOREM 28.36 (Relative isoperimetric inequality for the 2-ball). If E is ameasurable subset of B(r) c JR^2 and Area(E):::; ~7rr^2 , then
(28.35) PcEo(E, B(r))^2 ~ -Area(E).^8
7r
By Lemma 28.34, we have the following level set version of the above result.COROLLARY 28.37 (Level set version of the relative isoperimetric inequality
for the 2-ball). Suppose that f : B(r) ---+ [O, oo) is a smooth function satisfying
1l^1 (f-^1 (c)) < oo for all c E [O, oo). Then for each c E [O, oo) we have that
(28.36) (H^1 (f-^1 (c)))^2 ~ ~ min{H^2 (f-^1 [0, c) n B(r)), H^2 (f-^1 [c, oo) n B(r))}.
7r