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Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

enother hm sndwih in the plne
joint work with elexey flitskiy nd omn ursev from wosow snstitute of hysis nd ehnology elexey qrer
Moscow State University and Delone Lab oratory of Yaroslavl State University, Russia

¤ uolloquium uer uomintorikD slmenu xovemer WD PHIQ

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

rm sndwih theorem
heorem @o'eeErek versionA
Every sandwich with ham and cheese can b e cut by one planar cut in such way, that b oth pieces will contain equal amount of bread, ham, and cheese.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

rm sndwih theorem

heorem @mthemtil versionD due to tone nd ukey @IWRPA nd teinhus @IWRSAA
Every

d

nice measures in

R

d can b e cut by one hyp erplane in such

way, that each of two halfspaces will contain half of each measure.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

rm sndwih theorem

heorem @mthemtil versionD due to tone nd ukey @IWRPA nd teinhus @IWRSAA
Every

d

nice measures in

R

d can b e cut by one hyp erplane in such

way, that each of two halfspaces will contain half of each measure.

xie mesure µX mesure of the whole spe is I @or t lest (niteAY mesure of eh hyperplne is HF his property ould e omitted ut with dditionl restritionsF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

opologil nestors

heorem @forsukElm theoremA
For every continuous map
x

Sd

such that

f : Sd - R f (x) = f (-x).

d there exists a p oint

heorem @vyusternikEhnirelmn theoremA
If the sphere

Sd

is covered by

d +I

closed sets, then at least one of

these sets contains a pair of antip o dal p oints.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

opologil nestors

heorem @uker9s lemmA
Assume

T

is a triangulation of

d

-dimensional ball with all vertices

lab eled by numb ers from the set

{+I, -I, . . . , +d , -d }

. If on the

b oundary this lab eling is antip o dal (i.e. set of vertices on the b oundary is antip o dal and a opp osite p oints are lab eled with opp osite numb ers) then there is an edge with vertices lab eled with opp osite numb ers.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

opologil nestors

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

iquiprtitions with fns

he(nition e k -fan is olletion of k rys on the plne with ommon initil pointF heorem @f¡ ¡ D wtou§ PHHIA rny sekD
Any three measures can b e simultaneously halved by a divided into parts

P

-fan;

(3, 3)

2

1

by a

P-fan.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

iquiprtitions with fns
heorem @f¡ ¡ D wtou§ PHHIA rny sekD
Any two measures can b e divided into parts


1 2

and

I-

by a

P

-fan;

equipartitioned by a divided into parts divided into parts

Q

-fan; by a
1

1 (2, 4, 1) 4 1 1

Q

-fan;

(5, 5, 5, 5)

by a

R

-fan.

heorem @f¡ ¡ D wtou§ PHHPA rny sekD
Any two measures can b e divided into four equal parts each by a

R

-fan.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

iquiprtitions into onvex prts
gonjeture @unekoD unoD PHHPA
For any into

d

measures in

R

d and for any

k

the is a partition of
1

R

d

k

convex part such that each part contains

k of each measure.

¡ his onjeture ws independently proved y oeronD PHIHD nd ursevD PHIH heorem @ekopynD ursevD PHIQA
Under some additional restrictions for any one can cut the same fraction from all measures by a halfspace; cut the same prescrib ed fraction from all measures by a convex set.
A.Garb er Another ham sandwich in the plane MSU and Delone Lab of YSU

d +I

measures in

R

d


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

iquiprtitions into onvex prts

gonjeture @xndkumrD mno oD PHHVA
For any equal

n

a convex b o dy in

Rd

can b e cut into

d

-dimensional volume and equal

n convex parts of (d - I)-dimensional surface

volume.

eronovD rurdD nd ursev in PHIQ proved this onjeture for n equl to power of primeF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

iquiprtitions with polynomil surfes

heorem @qromovD PHHQA
If

nI

then every

n +d n

-I

measures in

R

d can b e divided into

halves by a p olynomial surface of degree at most

n

.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

§ olyhedrl urtin theorem y de ivljevi¡
gonsider d Esimplex = conv{x0 , . . . , in its interiorF he(nition e polyhedral curtain of is the one with vertex t O over join F1 F2 of oundries of two omplementry fes of F st is one over (d - P)Edimensionl polyhedrl sphereF
xd

} ontins the origin O

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

§ olyhedrl urtin theorem y de ivljevi¡
gonsider d Esimplex = conv{x0 , . . . , in its interiorF he(nition e polyhedral curtain of is the one with vertex t O over join F1 F2 of oundries of two omplementry fes of F st is one over (d - P)Edimensionl polyhedrl sphereF eh (d - P)Efe of se ontins ll verties of F1 exept one nd ll verties of F2 exept oneF
How to construct curtains: xd

} ontins the origin O

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

§ olyhedrl urtin theorem y de ivljevi¡
gonsider d Esimplex = conv{x0 , . . . , in its interiorF he(nition e polyhedral curtain of is the one with vertex t O over join F1 F2 of oundries of two omplementry fes of F st is one over (d - P)Edimensionl polyhedrl sphereF eh (d - P)Efe of se ontins ll verties of F1 exept one nd ll verties of F2 exept oneF
How to construct curtains: xd

} ontins the origin O

§ heorem @ivljevi¡ PHIQA D
For any

d

measures in

p olyhedral curtain of
A.Garb er Another ham sandwich in the plane

Rd there exist a that divides all

translation of some

measures into equal parts.
MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

hi'erene in twoE nd threeEdimensionl ses
R2
eny urtin is n ngle from feEfn of the simplex @plnr QEfnAF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

hi'erene in twoE nd threeEdimensionl ses
R2
eny urtin is n ngle from feEfn of the simplex @plnr QEfnAF

Rd with d Q
ome urtins re onstruted from severl ones from feE fn of the simplex

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

lnr generliztions of polyhedrl urtin theorem
heorem @flitskiyD eFqFD ursevD PHIQA
Any two measures on the plane can b e equipartitioned by a translation of some angle from an arbitrary

k

-fan for o dd

k

.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

lnr generliztions of polyhedrl urtin theorem
heorem @flitskiyD eFqFD ursevD PHIQA
Any two measures on the plane can b e equipartitioned by a translation of some angle from an arbitrary

k

-fan for o dd

k

.

heorem
Any two measures on the plane can b e equipartitioned by a translation of some angle from an arbitrary symmetric

Rk

-fan.

heorem
Any two measures on the plane can b e equipartitioned by a translation of some angle from an arbitrary

k

-fan if this fan contains

two opp osite rays with even numb er of angles b etween them.

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

sde of proof

gonsider red mesure

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

sde of proof

nd n ngleF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

sde of proof

hrw the set of verties of trnslted ngle tht ut the hlf of mesureF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

sde of proof

here re two rys tht we n see in this setF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

sde of proof

hese rys de(nes the red ngle orresponding to the initil red mesureF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

sde of proof

epet the sme onstrution for the lue mesureF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

sde of proof

sf there is no desired trnsltion of initil ngle then one of onstruted ngles ontins notherF

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

ome ounterexmplesX lmost ritrry REfn

F3

F

4

F

2

F

1

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

ome ounterexmplesX (Rk + P)Efn

F

1

F F
2

2k -1

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

ome ounterexmplesX Rk Efn

F

1

F2

k -1

F2

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU


Ham sandwich theorem

Measures equipartitions

Polyhedral curtain theorem

Equipartition with angle translations

THANK YOU!

A.Garb er Another ham sandwich in the plane

MSU and Delone Lab of YSU