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1. Changes in agglomeration and productivity are poor predictors of inequality across the archaeological record

   https://doi.org/10.1073/pnas.2400693122  

   Ortman, Scott G; Bogaard, Amy; Munson, Jessica; Lawrence, Dan; Green, Adam S; Feinman, Gary M; Chirikure, Shadreck; Uhl, Johannes H; Leyk, Stefan (April 2025, Proceedings of the National Academy of Sciences)

   We address three basic issues regarding the long-term dynamics of inequality in society. First, we consider the interpretation of residence sizes in socioeconomic terms by comparing statistical patterns extracted from the Global Dynamics of Inequality (GINI) Project database with those from the 21st-century United States. Second, we examine the degree to which the size and productivity of human networks is systematically related to inequality. Finally, we investigate relationships between productivity and productivity growth distributions for patterns of inequality and development across preindustrial societies. We find that across preindustrial societies residence size distributions provide a reasonable proxy for the distribution of productivity (income, a flow of physical and social resources to the group) and a minimum estimator for the distribution of wealth (a stock of such resources accumulated over time); that scale and productivity affect levels of inequality but account for only a small fraction of the observed variance across societies; and that inequality growth is independent of productivity growth, on average and over time. These findings have important implications for efforts to promote more equitable economic development in the present. 

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2. Proof of the Michael–Simon–Sobolev inequality using optimal transport

   https://doi.org/10.1515/crelle-2023-0054  

   Brendle, Simon; Eichmair, Michael (August 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2. 

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3. Affine isoperimetric inequalities for higher-order projection and centroid bodies

   https://doi.org/10.1007/s00208-025-03271-x  

   Haddad, Julián; Langharst, Dylan; Putterman, Eli; Roysdon, Michael; Ye, Deping (August 2025, Mathematische Annalen)

   Abstract In 1970, Schneider introduced the$$m$$ $m$th order difference body of a convex body, and also established the$$m$$ $m$th-order Rogers–Shephard inequality. In this paper, we extend this idea to the projection body, centroid body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang’s projection inequality, Petty’s projection inequality, the Busemann–Petty centroid inequality and Busemann’s random simplex inequality). We also establish a new proof of Schneider’s$$m$$ $m$th-order Rogers–Shephard inequality. As an application, a$$m$$ $m$th-order affine Sobolev inequality for functions of bounded variation is provided. 

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4. Urban form and structure explain variability in spatial inequality of property flood risk among US counties

   https://doi.org/10.1038/s43247-024-01337-3  

   Ma, Junwei; Mostafavi, Ali (December 2024, Communications Earth & Environment)

   Abstract Understanding the relationship between urban form and structure and spatial inequality of property flood risk has been a longstanding challenge in urban planning and emergency management. Here we explore eight urban form and structure features to explain variability in spatial inequality of property flood risk among 2567 US counties. Using datasets related to human mobility and facility distribution, we identify notable variation in spatial inequality of property flood risk, particularly in coastline and metropolitan counties. The results reveal variations in spatial inequality of property flood risk can be explained based on principal components of development density, economic activity, and centrality and segregation. The classification and regression tree model further demonstrates how these principal components interact and form pathways that explain spatial inequality of property flood risk. The findings underscore the critical role of urban planning in mitigating flood risk inequality, offering valuable insights for crafting integrated strategies as urbanization progresses. 

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5. Arakelov inequalities in higher dimensions

   https://doi.org/10.1515/crelle-2023-0075  

   Kovács, Sándor J; Taji, Behrouz (November 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over regular quasi-projective curves.We show that, for families of manifolds with ample canonical bundle, this invariant is uniformly bounded.As a consequence, we establish that such families over a base of arbitrary dimension satisfy the aforementioned Arakelov inequality, answering a question of Viehweg. 

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