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Unsolved problems on convex bodies posed by Busemann and Petty (1956)

Investigate and resolve the unsolved problems on convex bodies formulated by Herbert Busemann and C. M. Petty in their 1956 paper “Problems on convex bodies” (Math. Scand. 4), which concern geometric properties and measures of convex bodies in Euclidean spaces.

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Background

In 1956, Busemann and Petty published a list of research problems on convex bodies, spanning topics such as curvature, measures, and geometric inequalities in Euclidean spaces. These problems influenced subsequent work in convexity and geometry of normed spaces.

The article notes that many of these problems remain unresolved and references a commentary by D. Burago discussing their status and impact.

References

In 1956, together with C. M. Petty, Busemann formulated a set of problems on convex bodies, in a paper titled Problems on convex bodies [38]. Most of these problems are still open, see the commentary by D. Burago [14] in Volume II of Busemann’s Selected Works.

— Geometry in the twentieth century: A return to Euclid -- The work of Herbert Busemann (2406.01109 - Papadopoulos, 3 Jun 2024) in Section 6 (Convexity theory)