( Mj , g; , p; ): complete . Rig;] Z - in - is g; .

I B. Ip; 't z v > o

( M; . gj . p; )→ ( X

, dx . p ) .

pointed GH

Last time :. any tangent cone on X is a metric cone

.

( Y , dy , y ) is a tangent cone at x E X, if

F r. →o sit . ( Y , dy

, y ) = him IX. ri'dx, x )

• if IR"

is a tangent cone at x ,

then every tangent

cone at x = R"

.

RIX ) = { xt X : IR"

is a tangent cone at x }.

regular set .

3 ( x ) = X l RIX) singular set.

5k ( X ) = { x t X : no tangent cone at x splits offan Rk" factor }

3kg (X ) = { xe X : deft ( Brix) , B. Loh". Ei ) z y - r.

V re lo, i ] , ht cone Rk" x ( (Z ) }

( Y , dy ) : metric space .

supA E Y . It

"

( Al = YI, .

94ms IAI .

Hms ( A ) = int I won ram,

int is over coverings of balls A E Y Br.

lxa )

S.t. ta E S.

It"

( A ) s Hms IA ) e i FIT ( A ) .

¥general

dim#A = int { mzo : 94mA) = o }

=

sup { mzo : HMLA ) = is } .

St : " IA ) z ww s- P

. Hmu ) .

tem t xe X , f = dip , x) . V re R .

o e c. ( n . V , ft R ) r"

E Ot"

( B. txt ) E C,Cn ) r

"

.

⇒ dim X = n . & SIX) = 5""

( x) .

p - GHX; → x

,x; t Mj . Brix;) → Brlxl

( s ) E Cn - s-n

. rn.claim CME

.

it

Capa ( s ) =

"

Max # { Bs,.ly; ): { Bs, ly;) 's disjoint } .

Suppose N = Capping, Cs ) s small .

{13%1%1} disjoint in B-rlxjl .

Iommi > :÷Y÷, ". " "'

Z (fr )"

. I Brigit .

IBr I 2 II,

I Bsialyail ? N . IIT )"

. lBrh

def 're

⇒ It ! ( Tsrlxj ) E Cappy, Is) . wn 's"

E Cn . SJ . r"

. wns/ E Cn r"

.

GH

swktemma Y,→ T compact .

⇒ stmsHszm;IHIslYT;V Eso , E c s

Tf 3- Braly . ) , ra e s. T E Y 13,1gal .

sit .

It :( y ) + c z I um ram.

Br.ly: ) E Tj . T; E Y Baar.lyil .

Ss

Braly .) .for large j .

947,1T; ) e I walked"

ri ← ( Hsin { 94ms Hits }

s smallOn Mj . 113,1g > I E VI

, ( s ) =V 'll ' ) s

"

± Cns"

HN

Its l Brix; ' ) 7 Cn " I Brlxjll -

e - dip , ×,it Irl Ss

> C" T.FR#,/Be+r+,lxj/ .

dip; , xp .

-

Billy )✓ - r

"

7 Cu -VI

, let Rtl ) .

• 54×1 = Y.

571×1 .

RHS E LHS .

X t Jklx) l Y. 541×1 .

tf y > 0 , I try t ( o , I] sit.

dc://3.gl#iBzf.ry .

for'

some cone 112k" x ( ( Zg ) .

Case t : int rn = o -

§R! .az) .

po

rn,→ o - ( X , rn! dx . x ) → IT . dy , yl .

Case 2 : int ry > a > O u

y > o

Ba Ix ) E'Ba ( ok

"

,z't

) ( y -so I .

Y : metric space . M ? o . A E Y is 94M- measurable.

Then for 7 at A. Hms bins

m

him sup 9tolAnBr z z- m

,

r→ o Wm pm

If moreover 94mA ) co , then for 94M - a. e. a 1- A .

Lin supTtm ( An Balai)- E l .

t-so Wm rm

dim 5 ; IX) e k .

Lee ( X; , p ; ,"

( x. pl . Then

t.imsy.sk/xjlE5hylx)j -sis

& bing.INT/s4lxpnBilp;-IxBp)tf x; t 5 :( x; ) & x; → x ⇒ x t 541×1 .

✓

It C- lost)

suppose x # 541×1*1%1 Brix) , Balow's 2-* 1)①Yr< ( y - s ) r .

d :* ( Brlxjl . Brm ) s Ezr .

• dim 341×1 s k .

Suppose not . dim Sky IX) = m s k.

Hml 341×1 ) > o .

F X E 54 IX) w/ density > o ,i.e

.

limeys 9-151541×1 n Brix, )r-so- Z 2-

m

.

Wm pm

F ta → o,

' l

( Y , dy , y ) = big.

(X . ri'dx , x )

HII stfu)Wiz hinsy, gym

.I 841 ri'dx ) n Baha )

Iwm 2-m

.

dinghy ) = m .

1 lek .

Y = Rdx CIZI .I y , E 5514) n B. ly )

(T ,siay¥*µ,

Rxuz ,Y ' * Reisz- diy . .

Rex { E3 ) z '

z .

Blow up at ya .

Wussy. Hmu I 54141 n Brig . ) )- ? 2-

m

.

Wm pm

z ta → O ' ' R'" x ( ( z . )"

( Y , ri'

dy , y , ) → IT . . dy . .E )

Il

Rx UZI . tangent cone at y . .

PGHsplitting Thin : (MT . 5; -F;) → I w . dw

, Fl

Kc Fg;] Z - in - i ) S; .

S; →

o .

7 geodesic line on W ⇒ W = Rx W'

.

Repeat & stop at Thee , = Rk" x ClZ* ) .

9411 54 ( The . ) n 13,12*1 ) 7 wmz- m

- o .

-

" arbitrary

✓clan& 54 Ints - { new :

541-

) = f . do.µlBrlw) , Brlohtiz'T)Rhett> nr }

V r Elo, I ] , -V cone

( M; , g; , p;) → ( X , dx, p ) .

⇒ I B. lppl → 9-it Brlp ) .

Spherical .

51×1 = 5"

IX) .

C-

STIX) & 34 . . (x) = { xe X : data# IN . Blok", ziti ) z ysV S E Er, I ] , tf cone } .

It" ( Br Is! . Ix, ) n B. Ip , ) e cln.v.y.ir

"- h

tf S 70