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1. Exclusion volumes of convex bodies in high space dimensions: applications to virial coefficients and continuum percolation

   https://doi.org/10.1088/1742-5468/ac8c8b  

   Torquato, Salvatore; Jiao, Yang (September 2022, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume v ex ( K ) for a general convex body K that applies in any space dimension. While our main interests concern the rotationally-averaged exclusion volume of a convex body with respect to another convex body, we also describe some results for the exclusion volumes for convex bodies with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume v ex ( K )/ v ( K ) among all convex bodies, whether randomly oriented or uniformly oriented, for any d , where v ( K ) is the volume of K . When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any d with a large- d asymptotic scaling behavior of 2 2 d / d 3/2 , which is to be contrasted with the corresponding scaling of 2 d for the sphere. We present explicit formulas for quermassintegrals W 0 ( K ), …, W d ( K ) for many different nonspherical convex bodies, including cubes, parallelepipeds, regular simplices, cross-polytopes, cylinders, spherocylinders, ellipsoids as well as lower-dimensional bodies, such as hyperplates and line segments. These results are utilized to determine the rotationally-averaged exclusion volume v ex ( K ) for these convex-body shapes for dimensions 2 through 12. While the sphere is the shape possessing the minimal dimensionless exclusion volume, we show that, among the convex bodies considered that are sufficiently compact, the simplex possesses the maximal v ex ( K )/ v ( K ) with a scaling behavior of 2 1.6618… d . Subsequently, we apply these results to determine the corresponding second virial coefficient B 2 ( K ) of the aforementioned hard hyperparticles. Our results are also applied to compute estimates of the continuum percolation threshold η c derived previously by the authors for systems of identical overlapping convex bodies. We conjecture that overlapping spheres possess the maximal value of η c among all identical nonzero-volume convex overlapping bodies for d ⩾ 2, randomly or uniformly oriented, and that, among all identical, oriented nonzero-volume convex bodies, overlapping simplices have the minimal value of η c for d ⩾ 2. 

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2. Convexity of multiplicities of filtrations on local rings

   https://doi.org/10.1112/S0010437X23007972  

   Blum, Harold; Liu, Yuchen; Qi, Lu (April 2024, Compositio Mathematica)

   We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions. 

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3. Quantum algorithm for estimating volumes of convex bodies

   Chakrabarti, Shouvanik; Childs, Andrew; Hung, Shih-Han; Li, Tongyang; Wang, Chunhao; Wu, Xiaodi (January 2020, Annual Conference on Quantum Information Processing)

   Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an n-dimensional convex body within multiplicative error ϵ using Õ (n3+n2.5/ϵ) queries to a membership oracle and Õ (n5+n4.5/ϵ) additional arithmetic operations. For comparison, the best known classical algorithm uses Õ (n4+n3/ϵ2) queries and Õ (n6+n5/ϵ2) additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. 

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4. Using SOS and Sublevel Set Volume Minimization for Estimation of Forward Reachable Sets

   Jones, M.; Peet, M. (January 2019, IFAC proceedings series)

   In this paper we propose a convex Sum-of-Squares optimization problem for finding outer approximations of forward reachable sets for nonlinear uncertain Ordinary Differential Equations (ODE’s) with either (or both) L2 or point-wise bounded input disturbances. To make our approximations tight we seek to minimize the volume of our approximation set. Our approach to volume minimization is based on the use of a convex determinant-like objective function.We provide several numerical examples including the Lorenz system and the Van der Pol oscillator. 

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5. Quantum Algorithm for Estimating Volumes of Convex Bodies

   https://doi.org/10.1145/3588579  

   Chakrabarti, Shouvanik; Childs, Andrew M.; Hung, Shih-Han; Li, Tongyang; Wang, Chunhao; Wu, Xiaodi (September 2023, ACM Transactions on Quantum Computing)

   Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of ann-dimensional convex body within multiplicative error ε usingÕ(n³+ n^2.5/ε) queries to a membership oracle andÕ(n⁵+n^4.5/ε)additional arithmetic operations. For comparison, the best known classical algorithm usesÕ(n^3.5+n³/ε²)queries andÕ(n^5.5+n⁵/ε²)additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of “Chebyshev cooling,” where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requiresΩ (√ n+1/ε)quantum membership queries, which rules out the possibility of exponential quantum speedup innand shows optimality of our algorithm in 1/ε up to poly-logarithmic factors. 

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