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THE REFLEXIVE DIMENSION OF A LATTICE POLYTOPE

Combinatorics · math.CO · posed by Christian Haase, Ilarion V. Melnikov · open

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Statement

Is the reflexive dimension of the Minkowski sum P+P′P + P' bounded by refldim(P)+refldim(P′)+c\text{refldim}(P) + \text{refldim}(P') + c?

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  • THE REFLEXIVE DIMENSION OF A LATTICE POLYTOPE
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

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    Problem: Reconstructed statement: Does there exist a universal constant cc such that for all lattice polytopes P,P′P,P' in the same lattice,

    refldim⁡(P+P′)≤refldim⁡(P)+refldim⁡(P′)+c?\operatorname{refldim}(P+P')\le \operatorname{refldim}(P)+\operatorname{refldim}(P')+c?

    Here refldim⁡(P)\operatorname{refldim}(P) is the least dimension of a reflexive lattice polytope having a face lattice-equivalent to PP.

    Result: No. There is no such universal constant.

    Let M=Z2M=\mathbb Z^2, with basis e1,e2e_1,e_2. For N≥1N\ge 1, define

    PN=conv⁡(0,e1),PN′=conv⁡(0,e1+Ne2).P_N=\operatorname{conv}(0,e_1),\qquad P'_N=\operatorname{conv}(0,e_1+Ne_2).

    Both are primitive lattice segments, hence lattice-equivalent to [0,1][0,1]. The segment [0,1][0,1] has reflexive dimension 22: it is not a face of the unique one-dimensional reflexive segment, of lattice length 22, but it is an edge of the reflexive triangle

    conv⁡{(1,0),(0,1),(−1,−1)}.\operatorname{conv}\{(1,0),(0,1),(-1,-1)\}.

    Thus

    refldim⁡(PN)=refldim⁡(PN′)=2.\operatorname{refldim}(P_N)=\operatorname{refldim}(P'_N)=2.

    Their Minkowski sum is the parallelogram

    SN=PN+PN′=conv⁡(0,e1,e1+Ne2,2e1+Ne2),S_N=P_N+P'_N=\operatorname{conv}(0,e_1,e_1+Ne_2,2e_1+Ne_2),

    whose normalized lattice area is

    ∣det⁡(e1,e1+Ne2)∣=N.|\det(e_1,e_1+Ne_2)|=N.

    For each fixed DD, there are only finitely many DD-dimensional reflexive lattice polytopes up to lattice equivalence, and hence only finitely many two-dimensional faces among reflexive polytopes of dimension ≤D\le D. Therefore the normalized areas of all lattice polygons with reflexive dimension ≤D\le D are bounded by some constant ADA_D.

    Choosing N>ADN>A_D gives

    refldim⁡(SN)>D.\operatorname{refldim}(S_N)>D.

    Hence refldim⁡(PN+PN′)\operatorname{refldim}(P_N+P'_N) is unbounded while

    refldim⁡(PN)+refldim⁡(PN′)=4\operatorname{refldim}(P_N)+\operatorname{refldim}(P'_N)=4

    is constant. Therefore no universal additive constant cc can satisfy the proposed bound.

    Citation: Uses the standard finiteness theorem for lattice polytopes with one interior lattice point, in particular for reflexive polytopes; see Lagarias–Ziegler and the cited background in Haase–Melnikov, “The reflexive dimension of a lattice polytope,” arXiv:math/0406485.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The disproof is valid. The two summand segments are primitive, hence each has reflexive dimension 22. Their Minkowski sums are lattice parallelograms of unbounded normalized area NN. Since fixed reflexive dimension ≤D\le D permits only finitely many polygon faces up to lattice equivalence, their areas are bounded for fixed DD. Thus refldim⁡(PN+PN′)\operatorname{refldim}(P_N+P'_N) is unbounded while the summand reflexive dimensions stay 2+22+2, ruling out any universal additive constant cc.

    Novelty assessment

    TYPE1

    Classification rationale: The counterexample is genuinely simple: two primitive lattice segments have constant reflexive dimension, while their Minkowski sums give parallelograms of unbounded normalized area; fixed reflexive dimension bounds face types by finiteness of reflexive polytopes. This is a neat negative answer to the posed question, but it is an immediate short application of standard finiteness and is not substantial enough for a standalone combinatorics paper.

    Literature check: I found no prior resolution of the Minkowski-sum question. Searches for “reflexive dimension,” “refldim,” “reflexive dimension Minkowski sum,” the Haase–Melnikov title/authors, and arXiv:math/0406485 did not reveal a paper, note, forum post, or software discussion containing this counterexample or a stronger statement. OpenAlex lists the original arXiv preprint with only one citation and no apparent follow-up resolving this question.

    Citation: Original problem: Christian Haase and Ilarion V. Melnikov, “The reflexive dimension of a lattice polytope,” arXiv:math/0406485. Standard finiteness input: Lagarias–Ziegler, “Bounds for lattice polytopes containing a fixed number of interior points in a sublattice,” Canad. J. Math. 43 (1991), 1022–1035.

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