Finding the average is one of the key procedures in deteranalysis Yet littleattention is paid to why an arena

exists at all in the firstplace

Definition let X be a topological space Huu averaging is a ingpig associatingto a tupleits averageM l nuRequirements w se sc se

w is SnequivariantMkii Rin beta au foranypointsonhe is continuous

Topology of the Space Hi the ambo slackTate X S Then for fixed X sets walk aheaps 8 S There is whomof degreeof any cupping between

Compact oriented manifolds countingpreinases of a genniepants withorientation essentially looking at thereduced cuppingofHIX Hand IL the degree is an integer

So if the degree of mzCn S is D then

toy symmetry so is the degreeof C a S

However N 1 3 mix a se is

a cupping of degree 1 yet Xthe diagonal D flax C Sks

Data aggregation

is homologous to the sum of a parallel a meridian

on the torus six S so by funchialitythe degree of service a shouldbe equal to the sums of the degrees

of Rl wzCx 2 Khadr Ra I

i e QD Which cannot be L x

So i we 2 averaging on 8 for any sphereTurns out existence of h means for all u z means

that X is contractible

The B Eckman154 If X is homology equivalentto finite shephrideouplex or finitely way Suphies ineach dimension and there exist a averaging forany U2 2 then X is contractible

The proof is essentially taking the handgiesand establishing why the reasoning as above thatHulk 2 is h divisible for each h This is Kossthe onlyif Hulk 2 is like ie needs manyempties to

generate or is 0

Now if all homologies are zero then Hurumall hearehepie grepps one 0 which ihphies Whiteheadthat X is contractible

We should let that one can construct quiteearlySome pathological non contractible spaces wife wereprySuplest example is 2 solenoid defined as infinitesequences y ya Yu of points in Slsuch Thet ya 2yu for all u 4,2 Thenwz 41,42 n 41,42 1 42142,4343 1 is

a 2 averaging

Gaversely if X is www.acfible it admits n annoying

for all 2 This is proven by homotopy extension

easy but outside our scopeSeine examples are of course easy R has a whirl

averaging etcthere broad and igortarf class of examples is givenby

We war will be ceastruehry means require ware

from the space namely that X is a path metricspace will be assuming couplefeness I.e we havea dislaeae function dfa y and we can always

find a halfway point o

and therefore reconstruct a

geodesic the shortest timeconnecting a pair of points

Spaces of non-positive curvature.

Standard examples include of course Riemannian wfdsfar Ruder reefds

Among path metric spaces Nen Positive Curvature NPCspaces for any geoderie A

triangle ABC and ft c

D on the geodesic Be bB e

the distance e is E Dawhat it should love been C

1h Euclidean space a.k.a CATCo ar HadamardspearDowerfulcookeries as a distances in linear

beuotopies along

geodesics are convex

Distance is convex function in each argument

thetic balls are convex

i.e round spheres are not NPC

ds2 deedy7

Example Hyperbolic spaces

Iketric trees

Hilbert spaces duh but no further Banachspacesproduct of NPC spaces with d.sk do dsf

Gluing along convex subsets

Ihperlant properties of NPC spaces0

any two paints are connected by unique geodesichuegeodesic as a weaponry I 7 X depends continuouslyon its endpoints

BaycentersFor NPC spaces dYa is convex restriction to anygeodesic segment is convex

For any collection of points or better for anyprobability measure q with compact support defineits barycenter as the unique point bCq whichminimizes

a Jatta dqe

Banycenters define averaging on NPC spaces nMula Ru f Ta e easy to see it sahih

the axioms

Easy to see that for q a8a ta Ja PG is onthe geodesic sensed splitting it a ratio a i a Wewill call this point M no 2 ft t affaRemarkably simple algorithm to find beepFor any sequence of pants see au C Xwe define the doreded mean as

Mink Mz M 74,22142 MexFM Kc Mkl

I 1

Remark i then depends on order of pointsSampleHowever repeating

µwe have Ceuvergeue

The Sturm If m au isan iid Sample then 9 thenfun defined as above converges to b g

Application phylogenetic trees

Phylogenetic trees are metric trees with roots labeled

leaves same distance from roof to leaves

µ fiftystandard application deferreddistance

through mutationfrequencies

9 can be made into a metric1 3 2456

Space

HK 4

i z

XIf I473 Positions of the branches

parameterize cells cubes of cells is1 3 5 X X a 3

Theorem Biwa Holmes Westman spaceof phylogenetictrees is NPC

Droz Gromov if the link of any cube in a

cubical couplex is a flag complex then thespace is NPC Links in the spaceof treescorrespond to compatible famlies ofedges deputybested bisections they all can be variedsimultaneously a flag eenylex I