EEQUIPARTITION OF A SEGMENT
SERGEY AVVAKUMOV ♠ AND ROMAN KARASEV ♣ Abstract.
We prove that, for any positive integer m , a segment may be partitioned into m possibly degenerate or empty segments with equal values of a continuous function f of asegment, assuming that f may take positive and negative values, but its value on degenerateor empty segments is zero. Introduction
The mathematical theory of fair division develops along two main lines of research. On theone hand, it looks for partitions of the cake in m pieces (the positive integer m is fixed) ofa certain shape and equal in some sense, e.g. convex polygons of identical area. An earlyexample is the ham sandwich theorem [13, 14] about equipartitioning several measures byhyperplanes. More recent examples are Nandankumar’s conjecture [10] that every polygon canbe partitioned in m convex polygons of equal area and perimeter, solved in [1], and higher-dimensional analogues of Nandakumar’s conjecture that were solved in [4, 7] under assumptionthat m is a prime power. See also [12] for a result in between the ham sandwich theorem andNandakumar’s problem.On the other hand, the theory of fair division contributes key existence results to the conceptof fairness favored by economists and many social scientists, known as Envy Freeness . Forfurther references on it, see the founding work of Gale [6] and popular reviews [3, 15].Here we prove a topological property for partitions of a segment that we believe to be usefulto both types of results just mentioned. There is a single agent who evaluates each subsegment[ a, b ] of [0 ,
1] by a continuous utility function f ( a, b ) such that f ( a, a ) = 0 for all a . Apart fromthe continuity requirement, f is very general, in particular it can take both positive and negativevalues. We show the existence of an m -partition of [0 ,
1] in subsegments all of equal utility.This result does not add to the discussion of the general envy-free partitions of the segment,see also [11, 9, 2] for positive results on envy free divison of a segment under assumption that m is a prime power. But the following theorem is a key ingredient in the companion paper [5],where it implies the existence of a universal Fair Guarantee . Theorem 1.1.
Let I be the space of possibly degenerate subsegments [ a, b ] ⊆ [0 , . Assume wehave a continuous function f : I → R such that for degenerate segments we have f ([ a, a ]) ≡ for all a ∈ [0 , . Then for any positive integer m it is possible to partition the segment [0 , into m possibly degenerate segments [0 ,
1] = [0 , x ] ∪ [ x , x ] ∪ · · · ∪ [ x m − , so that f ([0 , x ]) = f ([ x , x ]) = · · · = f ([ x m − , . Mathematics Subject Classification.
Key words and phrases.
Fair partition, Configuration space. ♠ Supported by the European Research Council under the European Union’s Seventh Framework ProgrammeERC Grant agreement ERC StG 716424 – CASe. ♣ Supported by the Federal professorship program grant 1.456.2016/1.4 and the Russian Foundation forBasic Research grants 18-01-00036 and 19-01-00169. Each one of m agents compares pieces of the cake in her own way; we look for an m -partition where eachagent gets, in her own view, one of the best pieces. A utility level that can be achieved simultaneously by any m agents, each with her own utility function. a r X i v : . [ m a t h . M G ] S e p QUIPARTITION OF A SEGMENT 2
Let us first comment on the novelty of this result. This theorem looks like a particular case d = 1 of [1, Theorem 5.1], but it is not. The difference is that in Theorem 1.1 we also requirecertain behavior of f on degenerate segments and do not claim the validity of [1, Theorem 5.1]for d = 1 in the updated version of [1]. For d ≥
2, [1, Theorem 5.1] does not need such anassumption since the function of a convex body f in the proof is only applied to convex bodiesof a certain positive volume, thus excluding the need to consider degenerate parts.Let us also comment on the previously known particular cases. The case of non-negative f in Theorem 1.1 follows from the Knaster–Kuratowski–Mazurkiewicz theorem [8] in a standardway. The case of m a prime power and f of varying sign follows from the more general resultof [2]. The case of f additive on segments is an elementary exercise. Hence the new case hereis when the sign of f varies, f is not additive, and m is not a prime power. A generalization of Theorem 1.1 for envy free divisions, when m players divide a segmentinto m possibly empty parts and each of the players wants to receive one of the best partsaccording to his/her individual function f i : I → R , seems open for m not a prime power. Seefurther explanations and definitions on envy-free division of the segment in [11, 9, 2].The remarks that may only be understood after reading the proof are given in Section 3.2. Proof
The proof follows the argument in [1] and we are going to present it here. First, we passfrom single-valued functions to multi-valued functions. After rescaling we may assume that f in the statement of the theorem takes values in ( − , Definition 2.1. A nice multi-valued function I → [ − ,
1] is a compact subset Z ⊂ I × ( − , separates top from the bottom , that isthe sets I × {− } and I × { } belong to different connected components of I × [ − , \ Z .The following lemma allows us to consider nice multi-valued functions as ordinary continuousfunctions on the cylinder. Lemma 2.2. ( a ) For any nice multi-valued function of I , its graph is the zero set of an ordinarycontinuous function ϕ : I × [ − , → R such that ϕ ( I × {− } ) < and ϕ ( I × { } ) > . ( b ) For any ordinary continuous function ϕ : I × [ − , → R such that ϕ ( I × {− } ) < and ϕ ( I × { } ) > , its zero set is a graph of a nice multi-valued function.Proof. Claim (b) is trivial, so we prove (a). For a graph Z ⊂ I × ( − ,
1) of a nice multi-valuedfunction let ϕ be the signed distance to Z with a sign. It is possible to chose the sign arbitrarilyfor each connected component of I × [ − , \ Z ; any such signed distance function is continuous.The requirement for the sign of ϕ is achieved if we choose the sign of ϕ positive on the top andnegative on the bottom. (cid:3) In what follows we pass forth and back between the two points of view on multi-valued func-tions using Lemma 2.2. The function f from the statement of Theorem 1.1 can be consideredas a nice multi-valued function with ϕ ([ a, b ] , y ) = y − f ([ a, b ]). The boundary assumption f ([ a, a ]) ≡ a means that ϕ ([ a, a ] , y ) ≡ y for all a . Lemma 2.3 (A modification of Lemma 4.2 from [1]) . Assume ϕ : I × [ − , → R correspondsto a nice multi-valued function and ϕ ([ a, a ] , y ) ≡ y for all y . Let p be a prime. Then thereexists another nice multi-valued function ψ of I such that ψ ([ a, a ] , y ) ≡ y for all y and, whenever I ∈ I satisfies ψ ( I, y ) = 0 then there exists a partition I = I ∪ · · · ∪ I p into possibly degenerate segments such that (2.1) ϕ ( I , y ) = · · · = ϕ ( I p , y ) = 0 . QUIPARTITION OF A SEGMENT 3
Proof of Theorem 1.1 assuming the lemma.
Let m = p p . . . p n be a decomposition into primes.Let ϕ be the initial single-valued function f . Apply the lemma to ϕ and p to obtain ϕ . Thenapply the lemma to ϕ and p and so on. The final function ϕ n +1 will be a nice mutli-valuedfunction of a segment.From the definition it follows that a nice multi-valued assigns at least one value to anysegment. Hence there exists y ∈ ( − ,
1) such that ϕ n +1 ([0 , , y ) = 0 . In means that [0 ,
1] may be partitioned into p n possibly degenerate segments of the same value y of the multi-valued function ϕ n . Each of these segments may in turn be partitioned into p n − segments of the same value y of the multi-valued function ϕ n − , and so on. Eventually,we obtain a partition of [0 ,
1] into m = p · · · p n parts of the same value y of the multi-valuedfunction ϕ , which is in fact the single-valued function f . (cid:3) Proof of Lemma 2.3.
Parametrize the partitions of [ a, b ] ∈ I into p possibly degenerate seg-ments with the polyhedral model P p ⊂ F p ( R ) of the configuration space, introduced in [4](and denoted F (2 , p ) there). We briefly recall its properties: • P p has dimension p −
1, it is invariant with respect to the action of the permutationgroup S p on F p ( R ). • There is a single orbit of the top-dimensional cells of P p under the action of S p . Thetop-dimensional cells of P p may be oriented so that S p acts on these orientations by thesign of the permutation [4, Lemma 4.1]. • With these orientations, P p becomes a pseudomanifold modulo p [4, Section 4.2], thatis the homological boundary of thus constructed ( p − p .The parametrization of partitions with F p ( R ) or its subpolyhedron P p is defined as follows.Consider every point ( a , a ) ∈ R as a linear function (cid:96) : R → R , given by (cid:96) ( x ) = a x + a .The configuration space F p ( R ) is then considered as the space of p -tuples ξ = ( (cid:96) , . . . , (cid:96) p )of pairwise different linear functions. The partition of a segment I into possibly degeneratesegments I i is then defined as I i ( ξ ) = { x ∈ I | ∀ j (cid:54) = i (cid:96) i ( x ) ≤ (cid:96) j ( x ) } . The values ϕ ( I i ( ξ ) , y ) depend continuously on a configuration ξ ∈ F p ( R ) thanks to the as-sumption ϕ ([ a, a ] , y ) ≡ y for all y . Hence the equations(2.2) ϕ ( I ( ξ ) , y ) = · · · = ϕ ( I p ( ξ ) , y ) = 0 . then define a closed subset S ⊂ I × P p × [ − , a, b ] itself as thefirst variable of this product domain.Most of the proof below almost literally follows the proof of Lemma 4.2 in [1], but we give afull proof here for reader’s convenience.Let G p be the group of even permutations G p ⊂ S p for odd p . For p = 2, P is a circle withthe antipodal action of G := Z / Z . For odd p , P p of [4] is not that easy to describe.The set S is G p -invariant, where G p acts on P p as in [4] ( P p is a subset of the configurationspace of p -tuples of pairwise distinct points in R and G p permutes those points) and triviallyacts on I and [ − , S is the preimage of zero under the G p -equivariant continuous mapΦ : I × P p × [ − , → R p , Φ( I, ξ, y ) = ( ϕ ( I ( ξ ) , y ) , ϕ ( I ( ξ ) , y ) , . . . , ϕ ( I p ( ξ ) , y )) , where I i ( ξ ) denotes the i th part of the partition of I corresponding to the configuration ξ ∈ P p .This map is G p -equivariant if R p is acted on by G p by permutation of coordinates.Fix a segment I and study the structure of the fiber set S I = S ∩ ( { I } × P p × [ − , . QUIPARTITION OF A SEGMENT 4
When the G p -equivariant map Φ I = Φ | { I }× P p × [ − , is transverse to zero, the solution set S I is a finite number of points from the dimension con-siderations.If we make a homotopy of Φ I as a G p -equivariant map with the boundary conditions onits components ϕ ( I i , y ) then the solution set S I changes, but it changes in a definite way. Ifthe homotopy H : P p × [ − , × [0 , → R is transverse to zero (this can be achieved bya small perturbation) then H − (0) represents a G p -equivariant 1-dimensional cycle modulo p relative to P p × [ − , × { , } . Indeed, under the transversality assumption H − (0) consistsof smooth oriented segments in the top-dimensional faces of the domain P p × [ − , × [0 , p , the segments are attached to every point 0 modulo p times, unless we are at the boundary pieces P p × [ − , × { } and P p × [ − , × { } of thedomain, where the chain H − (0) has the boundary modulo p coinciding with the zero sets of theinitial Φ I ( · , · ) = H ( · , · ,
0) and the final Φ I ( · , · ) = H ( · , · , I changes equivariantly homologously to itself under G p -equivariant homotopies of themap Φ I .Let us present an instance of a transverse to zero map Φ : P p × [ − , → R p (a testmap), which is G p -equivariant and satisfies the boundary conditions that we impose on Φ I ,and for which the set Φ − (0) is homologically nontrivial. By the above homotopy consideration(connecting Φ to Φ I by convexly combining their coordinates), the existence of such a testmap implies the homological nontriviality of S I for any transverse to zero map Φ I . In order toproduce the needed test map, we may take the S p -equivariant test mapΨ : P p → W p considered in [4]. Here it is convenient to consider W p ⊂ R p as the linear subspace of p -tupleswith zero sum. The transverse preimage of zero Ψ − (0) consists of the unique S p -orbit of apoint in the relative interior of a top-dimensional face of P p . This solution set Ψ − (0) is eithera single G p -orbit (for p = 2) or splits into two G p -orbits (for odd p ).For p >
2, both G p -orbits of Ψ − (0) come with the same sign, because the permutation group S p ⊃ G p acts on the orientation of P p with the permutation sign [4, Lemma 4.1] and acts onthe orientation of W p with the permutation sign as well, thus proving that all points in the0-dimensional cycle Ψ − (0) must be assigned the same coefficient. This verifies the homologicalnontriviality of Ψ − (0) as a 0-dimensional G p -equivariant cycle.We augment Ψ to the map (assuming the coordinates of Ψ are in the interval ( − , ( ξ, y ) = Ψ ( ξ ) + ( y, . . . , y ) . Then Φ − (0) = Ψ − (0) × { } and this preimage is still a nontrivial G p -equivariant 0-cyclemodulo p . Hence, we obtain: Claim 2.4.
For transverse to zero Φ I , the set S I is a nontrivial G p -equivariant -cycle modulo p . Its projection to the segment [ − , is a nontrivial -cycle modulo p . Note that the set S I is always non-empty, since were it empty, the map Φ I would be transverseto zero by definition and S I would have to be non-empty by the claim. Assume now we changethe segment I in a continuous one-parameteric family { I ( s ) | s ∈ [ a, b ] } and obtain a G p -equivariant map with one more parameter (cid:101) Φ : P p × [ a, b ] × [ − , → R p , (cid:101) Φ( ξ, s, y ) = ( ϕ ( I ( ξ, s ) , y ) , ϕ ( I ( ξ, s ) , y ) , . . . , ϕ ( I ( ξ, s ) , y )) , where I i ( ξ, s ) is the i th part of the partition of I ( s ) corresponding to P p .The solution set (cid:101) Φ − (0) now generically (when (cid:101) Φ is transverse to zero) represents a G p -equivariant 1-dimensional cycle modulo p relative to P p × { a, b } × [ − , (cid:101) Φ − (0) consists of smooth oriented segments in QUIPARTITION OF A SEGMENT 5 the top-dimensional faces of the domain P p × [ a, b ] × [ − ,
1] and isolated points of intersectionwith the 1-codimensional skeleton of the domain; since the domain is a pseudomanifold modulo p , the segments are attached to every point 0 modulo p times, unless s = a or s = b .Projecting (cid:101) Φ − (0) to the rectangle [ a, b ] × [ − ,
1] (every G p -orbit goes to a single point), weget a 1-dimensional cycle modulo p relative to { a, b } × [ − , s = c nontrivially modulo p by the previous claim, since this is the solution set of a generic problemwithout a parameter. It is crucial that any curve connecting the bottom [ a, b ] × {− } to thetop [ a, b ] × { } of the rectangle is homologous to such a line, and it must intersect the cycle bythe homological invariance of the intersection number. Hence we obtain: Claim 2.5.
For a family of segments I ( s ) , the set S I ( s ) = S ∩ ( { I ( s ) | s ∈ [ a, b ] } × P p × [ − , separates top from the bottom when projected to [ a, b ] × [ − , . We have proved this for a transverse to zero (cid:101)
Φ, but the transversality assumption is notnecessary. Once we have a curve from [ a, b ] × {− } to [ a, b ] × { } not touching the projection ofthe solution set for an arbitrary G p -equivariant (cid:101) Φ, satisfying the boundary conditions, this curvewill not touch the projection of the solution set for a small generic (and therefore transverse tozero) perturbation of (cid:101)
Φ; but the latter is already shown to be impossible.Now we consider the cylinder
I × [ − , γ : [ a, b ] → I × [ − , I × {− } to the top I × { } in the cylinder and parametrized by asegment [ a, b ]. Its first coordinate may be considered as a one-parametric family of segments I ( s ). Hence applying the previous claim, we obtain: Claim 2.6.
The projection Z of S to I × [ − , separates the top I × { } from the bottom I × {− } . This is the crucial separation property of Z ⊂ I × [ − , ψ : I × [ − , → R may be obtained as a signed distance function. But we need to ensure that ψ ([ a, a ] , y ) ≡ y forall y , which may be not true for the signed distance function obtained from the proof of Lemma2.2. We are going to fix it, the following ending of the proof is different from the argument inthe proof of [1, Lemma 4.2].Put for brevity I (cid:48) = { [ a, a ] | a ∈ [0 , } . Examining our construction of S and Z in case ofa degenerate segment (which may only be partitioned into degenerate segments) and using thefact that ϕ ([ a, a ] , y ) ≡ y for all y , we see that(2.3) Z ∩ ( I (cid:48) × R ) = I (cid:48) × { } In order to obtain the property ψ ([ a, a ] , y ) ≡ y , we use a modification of the argument in theproof of Lemma 2.2 to build the function ψ : I × [ − , → R with zero set Z . Define ψ on Z as zero, on the set I (cid:48) × [ − ,
1] as y , and define ψ ( I, ≡ ψ ( I, − ≡ −
1. After this andbecause of (2.3) ψ is continuously defined on the closed set Y = Z ∪ ( I (cid:48) × [ − , ∪ ( I ×{− , } ) ⊇ Z .Then we extend ψ by the Titze theorem to the connected components of I × ( − , \ Y so that on the connected components touching the top it remains non-negative. By addingto ψ the distance function to the set Y in such components (and still denoting the resultingfunction by ψ ) we make ψ strictly positive in top components of the complement of Y . We do QUIPARTITION OF A SEGMENT 6 the same on the components of the complement of Y touching the bottom with the minus sign,thus extending ψ to a negative function there. In effect, we obtain ψ with zero set Z satisfying ψ ( I (cid:48) , y ) ≡ y , ψ ( I , >
0, and ψ ( I , − < ψ ( I, y ) = 0, the pair (
I, y ) is in Z and correspondsto ( I, ξ, y ) ∈ S . The latter triple, in turn, provides a partition of I into p segments I , . . . , I p satisfying ϕ ( I , y ) = · · · = ϕ ( I p , y ) . (cid:3) Final remarks
The parametrization of partitions of [0 ,
1] in m possibly degenerate or empty segments bythe configuration space F m ( R ) allows to reprove the positive envy-free division result from[2]. The assumption that the preference of a degenerate segment in a partition does not dependon the degenerate segment and is the same as the preference of the empty segment allows todefine closed subsets A ij ⊆ F m ( R ), where the player j agrees to take the part number i . Inorder to assume those sets closed, one really needs the assumption on degenerate and emptysegments, because in the parametrization by the configuration space degenerate segments maybecome empty segments after an arbitrarily small perturbation of the configuration of linearfunctions. This collection of sets agrees with the natural action of the permutation group onthe configuration space and its action by permuting the index i of the sets A ij .After that, approximating the coverings (cid:83) i A ij = F m ( R ) with partitions of unity and usingthe Birkhoff–von Neumann theorem as in [6] (see also [2, Theorem 2.2]), one builds an equi-variant map F m ( R ) → R m , such that an envy-free division exists when this map touches thediagonal of R m . For m a prime power, such an equivariant map must touch the diagonal bythe results of [7, 4].For m not a prime power, such an equivariant map F m ( R ) → R m need not touch the diagonalby the negative result of [4], and implicitly, by the negative result of [2]. So far we have no ideaif it is possible to deduce the negative result of [2] from the existence of an equivariant map F m ( R ) → R m . References [1] A. Akopyan, S. Avvakumov, and R. Karasev. Convex fair partitions into an arbitrary number of pieces.2018. arXiv:1804.03057.[2] S. Avvakumov and R. Karasev. Envy-free division using mapping degree. 2019. arXiv:1907.11183.[3] J. B. Barbanel, S. J. Brams, and W. Stromquist. Cutting a pie is not a piece of cake.
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Sergey Avvakumov, Department of Mathematical Sciences, University of Copenhagen, Uni-versitetspark 5, 2100 Copenhagen, Denmark
E-mail address : [email protected] Roman Karasev, Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgo-prudny, Russia 141700Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow,Russia 127994
E-mail address : r n [email protected] URL ::