Convex Geometry and Inequalities in Euclidean Spaces

Convex geometry studies the structural and metric properties of convex sets in Euclidean spaces, with an emphasis on how volume, surface area and other geometric measures interact under elementary operations. Central themes include the behaviour of convex bodies under Minkowski addition, the role of support functions in encoding boundary information, and the interplay between probabilistic and geometric inequalities. The Brunn–Minkowski inequality, which relates the volumes of two sets to the volume of their Minkowski sum, underpins many later developments, including functional forms of the inequality and extensions to log-concave measures. The classical Minkowski problem and its Lp variants seek to reconstruct a convex body from prescribed curvature or surface area measures, revealing deep links between partial differential equations and geometric analysis. Recent progress in high-dimensional convexity has addressed long-standing conjectures concerning isoperimetric coefficients in the Kannan–Lovász–Simonovits framework and sharpened stability estimates for volume inequalities. These advances have significant implications for areas as diverse as functional analysis, probability theory and algorithmic sampling methods.

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Convex Geometry and Inequalities in Euclidean Spaces publication trend

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Technical terms

Convex body: a compact convex subset of a Euclidean space with non-empty interior.

Minkowski sum: the set obtained by adding every vector in one convex body to every vector in another.

Support function: a map assigning to each direction the maximal distance from the origin to the supporting hyperplane of a convex body.

Brunn–Minkowski inequality: a foundational relation that compares the volumes of two sets to the volume of their Minkowski sum, asserting concavity of volume to the power 1/n.

Lp-Minkowski problem: the challenge of reconstructing a convex body whose generalized surface area measure equals a given measure raised to the pth power.

Isoperimetric coefficient: a quantitative measure of how surface area minimises relative to volume within a family of convex bodies.

Log-concave measure: a measure whose density function is log-concave, often yielding strong concentration and functional inequality properties.

References

1. The L p -Minkowski problem for −n . Advances in Mathematics (2019).
2. An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture. Geometric and Functional Analysis (2021).
3. Gaussian Brunn-Minkowski inequalities. Transactions of the American Mathematical Society (2010).