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1. Fooling Polytopes

   https://doi.org/10.1145/3460532  

   O’Donnell, Ryan; Servedio, Rocco A.; Tan, Li-Yang (April 2022, Journal of the ACM)

   We give a pseudorandom generator that fools m -facet polytopes over {0, 1} n with seed length polylog( m ) · log n . The previous best seed length had superlinear dependence on m . 

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2. Gelfand--Tsetlin Polytopes: A Story of Flow and Order Polytopes

   https://doi.org/10.1137/19M1251242  

   Liu, Ricky I.; Mészáros, Karola; Dizier, Avery St. (January 2019, SIAM Journal on Discrete Mathematics)
3. Alcoved Polytopes II

   https://doi.org/10.1007/978-3-030-02191-7_10  

   Lam, Thomas; Postnikov, Alexander (January 2018, Springer International Publishing)
4. Generalized Parking Function Polytopes

   https://doi.org/10.1007/s00026-023-00671-1  

   Hanada, Mitsuki; Lentfer, John; Vindas-Meléndez, Andrés R (June 2024, Annals of Combinatorics)

   Abstract A classical parking function of lengthnis a list of positive integers$$(a_1, a_2, \ldots , a_n)$$ $(a_1,a_2,\dots ,a_n)$ whose nondecreasing rearrangement$$b_1 \le b_2 \le \cdots \le b_n$$ $b_1\le b_2\le \cdots \le b_n$ satisfies$$b_i \le i$$ $b_i\le i$ . The convex hull of all parking functions of lengthnis ann-dimensional polytope in$${\mathbb {R}}^n$$ $R^n$ , which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of$${\textbf{x}}$$ $x$ -parking functions for$${\textbf{x}}=(a,b,\dots ,b)$$ $x=(a,b,\cdots ,b)$ , which we refer to as$${\textbf{x}}$$ $x$ -parking function polytopes. We explore connections between these$${\textbf{x}}$$ $x$ -parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of$${\textbf{x}}$$ $x$ -parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary. 

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5. Associahedra as moment polytopes

   https://doi.org/10.1112/blms.13212  

   Gekhtman, Michael; Thomas, Hugh (February 2025, Bulletin of the London Mathematical Society)

   Abstract Generalized associahedra are a well‐studied family of polytopes associated with a finite‐type cluster algebra and a choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani‐Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster ‐variety with universal coefficients by its maximal natural torus action. We prove our result by showing that the construction of Padrol, Palu, Pilaud, and Plamondon can be understood on the basis of the way that moment polytopes behave under symplectic reduction. 

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