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1. 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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2. 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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3. 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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4. The Global Dynamics of Inequality (GINI) project: analysing archaeological housing data

   https://doi.org/10.15184/aqy.2023.188  

   Bogaard, Amy; Ortman, Scott; Birch, Jennifer; Cervantes_Quequezana, Gabriela; Chirikure, Shadreck; Crema, Enrico R; Cruz, Pablo; Feinman, Gary; Fochesato, Mattia; Green, Adam S; et al (February 2024, Antiquity)

   Abstract The GINI project investigates the dynamics of inequality among populations over the long term by synthesising global archaeological housing data. This project brings archaeologists together from around the world to assess hypotheses concerning the causes and consequences of inequality that are of relevance to contemporary societies globally. 

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5. Frontier workers and the seedbeds of inequality and prosperity

   https://doi.org/10.1093/jeg/lbad018  

   Connor, Dylan_Shane; Kemeny, Tom; Storper, Michael (August 2023, Journal of Economic Geography)

   Abstract This article examines the role of work at the cutting of technological change—frontier work—as a driver of prosperity and spatial income inequality. Using new methods and data, we analyze the geography and incomes of frontier workers from 1880 to 2019. Initially, frontier work is concentrated in a set of ‘seedbed’ locations, contributing to rising spatial inequality through powerful localized wage premiums. As technologies mature, the economic distinctiveness of frontier work diminishes, as ultimately happened to cities like Manchester and Detroit. Our work uncovers a plausible general origin story of the unfolding of spatial income inequality. 

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