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A gallery of open problems in geometry that keep me puzzled and amused

Anton Petrunin

TL;DR

This collection assembles 18 open geometric questions spanning topics from two-convexity and braid spaces to CAT(0) geometry, biquotients, and polyhedral metrics on complex projective spaces. Each entry states a concrete puzzle, often accompanied by partial results, related conjectures, and connections to classical problems such as the Danzer–Grünbaum bound and AKP-invitation constructions, highlighting how curvature, symmetry, and metric properties interact in modern geometry. The compilation aims to stimulate targeted research by presenting precise, testable questions with implications across metric geometry, geometric topology, and geometric analysis. The overarching theme is to probe the limits of curvature-based methods, geometric group actions, and polyhedral approximations in diverse geometric contexts, potentially revealing new structural phenomena in high-dimensional spaces.

Abstract

This is a collection of open problems in geometry that I think of as puzzles: they stick to my brain -- I see many grips, but no spare hands. Puzzle-charm is the only criterion for including a problem here; importance is ignored.

A gallery of open problems in geometry that keep me puzzled and amused

TL;DR

This collection assembles 18 open geometric questions spanning topics from two-convexity and braid spaces to CAT(0) geometry, biquotients, and polyhedral metrics on complex projective spaces. Each entry states a concrete puzzle, often accompanied by partial results, related conjectures, and connections to classical problems such as the Danzer–Grünbaum bound and AKP-invitation constructions, highlighting how curvature, symmetry, and metric properties interact in modern geometry. The compilation aims to stimulate targeted research by presenting precise, testable questions with implications across metric geometry, geometric topology, and geometric analysis. The overarching theme is to probe the limits of curvature-based methods, geometric group actions, and polyhedral approximations in diverse geometric contexts, potentially revealing new structural phenomena in high-dimensional spaces.

Abstract

This is a collection of open problems in geometry that I think of as puzzles: they stick to my brain -- I see many grips, but no spare hands. Puzzle-charm is the only criterion for including a problem here; importance is ignored.

Paper Structure

This paper contains 18 sections, 8 equations, 1 figure.

Table of Contents

  1. Two-convexity
  2. Braid space
  3. Quotients of Hilbert space
  4. Nested convex surfaces
  5. Maximal finite subgroups
  6. Shortcut in a connected set
  7. Involution of the 3-sphere
  8. Faceting Fubini--Study
  9. Octahedron comparison
  10. Group generated by central symmetries
  11. Milnor's cartography problem
  12. Convex hull in CAT(0)
  13. Homeomorphism of the torus
  14. Open projection
  15. Zero-curvature along geodesics
  16. ...and 3 more sections

Figures (1)

  • Figure :