This volume grew out of the Fall 2023 special semester Discrete Geometry and Convexity at the Erdős Center of the Alfréd Rényi Institute in Budapest, where leading experts and outstanding young researchers gathered for an intensive program of mini-courses, workshops, and conferences. New Probes into Discrete and Convex Geometry brings together 17 invited survey articles that map the current landscape of the field and highlight some of its most dynamic frontiers. The chapters range from convexity and packing problems to combinatorial and piecewise-linear topology, from Helly-Tverberg theory and its topological methods to questions motivated by computer science, data analysis, and geometric optimization. Along the way, readers will find both polished expositions of major tools and guided tours of active research areas, rich with open problems and new perspectives. Written by world-class researchers, these surveys offer an accessible entry point for graduate students and newcomers, while providing specialists with a concise reference to recent advances.

János Pach is a research professor at the Alfréd Rényi Institute, Budapest. His main fields of interest are discrete and computational geometry, convexity, and combinatorics. He has written more than 350 research papers. His books, "Research Problems in Discrete Geometry'' (with Brass and Moser) and "Combinatorial Geometry" (with Agarwal) were translated into Japanese, Russian, and Chinese. He is the co-editor-in-chief of Discrete & Computational Geometry. He received the Lester Ford Award from the Mathematical Association of America (1990), the Rényi Prize and the Academy Award from the Hungarian Academy of Sciences (1993,1998), and the Szele Prize from the Bolyai Mathematical Society (2019). He was elected the ACM Fellow (2011) and was a member of Academia Europaea (2014) and the Hungarian Academy of Sciences (2022). He was an invited speaker at the International Congress of Mathematicians (2014) and a plenary speaker at the European Congress of Mathematics (2021).

Géza Tóth is a research professor and the head of the Department of Geometry at the Alfréd Rényi Institute, Budapest, working on problems in discrete and computational geometry and combinatorics. He received his PhD degree at Courant Institute, New York University in 1997, as a student of János Pach, winning the Best Dissertation Prize in mathematics and the Sokol Award, the highest university-wide prize for fresh PhDs. He has written more than 100 research articles and he is the co-editor-in-chief of Studia Scientiarum Mathematicarum Hungarica and the editor of Computational Geometry and Acta Mathematica Hungarica. Among other distinctions, he received the Erdős Prize of the Hungarian Academy of Sciences in 2008 and the Rényi Prize of the Rényi Institute in 2010.