This book reflects general trends in the study of geometric aspects of functional analysis, understood in a broad sense. A classical theme in the local theory of Banach spaces is the study of probability measures in high dimension and the concentration of measure phenomenon. Here this phenomenon is approached from different angles, including through analysis on the Hamming cube, and via quantitative estimates in the Central Limit Theorem under thin-shell and related assumptions. Classical convexity theory plays a central role in this volume, as well as the study of geometric inequalities. These inequalities, which are somewhat in spirit of the Brunn-Minkowski inequality, in turn shed light on convexity and on the geometry of Euclidean space. Probability measures with convexity or curvature properties, such as log-concave distributions, occupy an equally central role and arise in the study of Gaussian measures and non-trivial properties of the heat flow in Euclidean spaces. Also discussed are interactions of this circle of ideas with linear programming and sampling algorithms, including the solution of a question in online learning algorithms using a classical convexity construction from the 19th century.

高次元幾何学という広大な宇宙を舞台に、関数解析の深淵へ挑む珠玉の論考集です。測度の集中現象という「奇跡」を解き明かす筆致は、世界の真理を暴き出す推理小説のような緊迫感に満ちています。19世紀の凸体理論が、最先端の学習アルゴリズムと共鳴する瞬間のダイナミズムは、まさに知性の極致と言えるでしょう。 本書が描くのは、数式という言語で綴られた「空間の詩学」です。幾何学的不等式の美しさが、高次元の混沌に鮮やかな秩序をもたらす様は、読者の知的好奇心を激しく揺さぶります。数学という学問が持つ、時代を超越した生命力と圧倒的な美意識を、ぜひその手で確かめてください。