2024/12/31 by Takashi Hirotsu, Hirotsu, Takashi · 1 citation
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paper · pdf · doi:10.48550/arxiv.2501.00459
Let d ≥ 0 be an integer and let P ⊂ \mathbb Rd be a d-dimensional lattice polytope. We call a polytope M ⊂ \mathbb Rd such that M ⊂ P and M ∼ P a miniature of P, and it is said to be horizontal if M is transformed into P by translating and positive integral rescaling. A miniature M of P is said to be average-sized (resp. normal-sized) if the volume of M is equal to the limit of the sequence whose n-th term is the average of the volumes of all miniarures (resp. all horizontal miniatures) whose vertices belong to (n-1\mathbb Z)d. We prove that, for any lattice square P ⊂ \mathbb R2, the ratio of the areas of an average-sized miniature of P and P is 2:15. We also prove that, for any lattice simplex P ⊂ \mathbb Rd, the ratio of the volume of a normal-sized miniature of P to that of P is 1:\binom2d+1d. This ratio is same as the known result for the hypercube [0,1]d provided by the author.