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1. Minkowski vacuum entanglement and accelerated oscillator chains

   https://doi.org/10.1103/PhysRevD.111.045022  

   Svidzinsky, Anatoly A; Scully, Marlan O; Unruh, William (February 2025, Physical Review D)

   Minkowski vacuum is empty from the perspective of Unruh-Minkowski photons, however, in the Rindler picture, it is filled with entangled pairs of Rindler photons. A ground-state atom uniformly accelerated through Minkowski vacuum can become excited by absorbing a Rindler photon (Unruh effect) or, in the alternative description, by emitting an Unruh-Minkowski photon (Unruh-Wald effect). We find an exact solution for the quantum evolution of a long chain of harmonic oscillators accelerated through Minkowski vacuum and for two chains accelerated in the opposite directions. We show how entanglement of Rindler photons present in Minkowski vacuum is transferred to the oscillators moving in causally disconnected regions. We also show that in the Unruh-Minkowski photon picture the process can be interpreted as if initial correlations between collective oscillator modes are transferred to the generated Unruh-Minkowski photons. 

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2. Minkowski problems for geometric measures

   https://doi.org/10.1090/bull/1858  

   Huang, Yong; Yang, Deane; Zhang, Gaoyong (July 2025, Bulletin of the American Mathematical Society)

   This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and new Minkowski problems studied in convex geometry, PDEs, and harmonic analysis, and structured in a conceptual framework of the Brunn-Minkowski theory, its extensions, and related subjects. 

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3. The L _p chord Minkowski problem

   https://doi.org/10.1515/ans-2022-0041  

   Xi, Dongmeng; Yang, Deane; Zhang, Gaoyong; Zhao, Yiming (January 2023, Advanced Nonlinear Studies)

   Abstract Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n {{\mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p {L}_{p} chord measures is called the L p {L}_{p} chord Minkowski problem in the L p {L}_{p} Brunn-Minkowski theory, which includes the L p {L}_{p} Minkowski problem as a special case. This article solves the L p {L}_{p} chord Minkowski problem when p > 1 p\gt 1 and the symmetric case of 0 < p < 1 0\lt p\lt 1 . 

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4. Interpolation, Prekopa and Brunn-Minkowski for F-subharmonicity

   https://doi.org/10.1016/j.aim.2023.109405  

   Ross, Julius; Witt_Nyström, David (January 2024, Advances in Mathematics)

   We extend Prekopa’s Theorem and the Brunn-Minkowski Theo- rem from convexity to F-subharmonicity. We apply this to the interpolation problem of convex functions and convex sets introducing a new notion of “har- monic interpolation” that we view as a generalization of Minkowski-addition. 

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5. Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

   https://doi.org/10.1109/TCAD.2022.3197971  

   Yang, Liren; Zhang, Hang; Jeannin, Jean-Baptiste; Ozay, Necmiye (August 2022, IEEE transactions on computeraided design of integrated circuits and systems)

   Backward reachability analysis is essential to synthesizing controllers that ensure the correctness of closed-loop systems. This paper is concerned with developing scalable algorithms that under-approximate the backward reachable sets, for discrete-time uncertain linear and nonlinear systems. Our algorithm sequentially linearizes the dynamics, and uses constrained zonotopes for set representation and computation. The main technical ingredient of our algorithm is an efficient way to under-approximate the Minkowski difference between a constrained zonotopic minuend and a zonotopic subtrahend, which consists of all possible values of the uncertainties and the linearization error. This Minkowski difference needs to be represented as a constrained zonotope to enable subsequent computation, but, as we show, it is impossible to find a polynomial-size representation for it in polynomial time. Our algorithm finds a polynomial-size under-approximation in polynomial time. We further analyze the conservatism of this under-approximation technique, and show that it is exact under some conditions. Based on the developed Minkowski difference technique, we detail two backward reachable set computation algorithms to control the linearization error and incorporate nonconvex state constraints. Several examples illustrate the effectiveness of our algorithms. 

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