Barycenters of Points That Are Constrained to a Polytope Skeleton

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Theorem 5. Let n be prime, d ≥ 1 be an integer, and F ⊆ Rd be closed. If there is an (n − 2)-connected, S ⊕d Rd×n n-invariant subset Q ⊆ Wn = {(x1,...,xn) ∈ | P xi =0} such that for each (x1,...,xn) ∈ Q there is an i with xi ∈ F , then there are p1,...,pn ∈ F with p1 + ··· + pn =0.

n Proof. The map Ψ: Q → R , (x1,...,xn) 7→ (dist(x1, F ),..., dist(xn, F )) is Sn-equivariant. Denote the n diagonal by D := {(y1,...,yn) ∈ R | y1 = ··· = yn}. The map Ψ induces by projection an Sn-equivariant ⊥ map Φ: Q → D = Wn. The vector space Wn is (n − 1)-dimensional and Q is (n − 2)-connected, so Φ has a zero by Lemma 4 applied to the subgroup Z/n of Sn, which acts freely on Wn \{0}. Let (p1,...,pn) ∈ Q with Φ(p1,...,pn)=0. This is equivalent to dist(p1, F ) = ··· = dist(pn, F ). There is an index i such that ⊕d dist(pi, F )=0, or equivalently pi ∈ F , since F is closed. Thus all pj are in F . Since Q ⊆ Wn we have p1 + ··· + pn =0.

This theorem can be extended to the case that n is a prime power using a generalization of Dold’s theorem to elementary Abelian groups; for a rather general version see [3]. For the proof of Theorem 2 we take for Q a skeleton of a certain polytope. Before proceeding we point out that the following naïve approach using a conﬁguration space/test map scheme to prove Theorem 1 fails: the origin (d) n dn is the barycenter of n points in the d-skeleton of P if and only if the Sn-equivariant map (P ) → R , 1 (d) n (x1,...,xn) 7→ n (x1 + ··· + xn) maps some point in (P ) to 0. However, for this map to be equivariant the dn (d) n dn Sn-action on R must be trivial, and thus Sn-equivariant maps (P ) → R \{0} exist. Dobbins’s novel idea was to intersect with a test space in the domain to avoid this problem. We will employ that same idea. In contrast to Dobbins we need no requirement of general position. Thus we also do not need any kind of approximation in the proof.

Proof of Theorem 2. We can assume that p = 0 is in the interior of P , otherwise we could restrict to a proper ⊕d face of P with the origin in its relative interior. Let ﬁrst n be prime. Consider again the linear space Wn = Rd×n n ⊕d {(x1,...,xn) ∈ | P xi = 0} of codimension d. Then C := P ∩ Wn is a polytope of dimension (n − 1)d. The (n − 1)-skeleton C(n−1) of C is homotopy equivalent to a wedge of (n − 1)-spheres and thus is (n − 2)-connected. (n−1) (k) Let (x1,...,xn) ∈ C . By Theorem5 we need to show that one xi lies in P . Suppose for contradiction (k) that xi ∈/ P for all i = 1,...,n. For each xi let σi be the inclusion-minimal face of P with xi ∈ σi. We ⊕d have that dim σi ≥ k + 1. Each face of C is of the form (τ1 ×···× τn) ∩ Wn with the τi faces of P . ⊕d The point (x1,...,xn) lies in the face (σ1 ×···× σn) ∩ Wn but in no proper subface. The dimension is ⊕d ⊕d (n−1) dim (σ1 ×···× σn) ∩ Wn ≥ n(k + 1) − d ≥ n. Thus (σ1 ×···× σn) ∩ Wn ∈/ C , which is a contradiction. The case for general n follows by a simple iteration with respect to prime divisors, as in [4].

References

[1] Luis Barba et al., Weight Balancing on Boundaries and Skeletons, 30th Symp. Comp. Geom., 2014, pp. 436–443. [2] Pavle V. M. Blagojevic,´ Florian Frick, and Günter M. Ziegler, Tverberg plus constraints, Bull. Lond. Math. Soc. 46 (2014), no. 5, 953–967. [3] Pavle V. M. Blagojevic,´ Wolfgang Lück, and Günter M. Ziegler, Equivariant topology of conﬁguration spaces, arXiv:1207.2852 (2012), to appear in J. Topology. [4] Michael G. Dobbins, A point in a nd-polytope is the barycenter of n points in its d-faces, Invent. Math. (2014), 6 pages, doi: 10.1007/s00222-014-0523-2. [5] Albrecht Dold, Simple proofs of some Borsuk–Ulam results, Contemp. Math. 19 (1983), 65–69. [6] Jiríˇ Matoušek, Using the Borsuk–Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry, second ed., Universitext, Springer-Verlag, Heidelberg, 2008.