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Cartan–Hadamard isoperimetry

Geodesic-ball volume comparison does not imply boundary-area comparison for arbitrary domains.

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Understand the problem ↓

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Understanding the problem

Read the background

Geodesic-ball volume comparison does not imply boundary-area comparison for arbitrary domains. The target is the full isoperimetric inequality, beyond known dimensions and special classes.

The curvature, completeness, and simple-connectivity hypotheses are retained. Euclidean equality rigidity is stated for bounded positive-volume sets; no unrestricted equality classification is inferred.

Why it matters

If correct, extends the stated isoperimetric comparison to all dimensions and arbitrary finite-perimeter sets under its hypotheses.

What counts as progress

A sourced resolution or substantial advance on the precise target, with hypotheses, prior context, and verification evidence stated separately.

What to watch for

  • Assess the precise statement and assumptions before comparing results.
  • The curvature, completeness, and simple-connectivity hypotheses are retained. Euclidean equality rigidity is stated for bounded positive-volume sets; no unrestricted equality classification is inferred.
  • Track independent assessments, dependencies, corrections, and withdrawals separately from importance.

Background review: 2026-10-11. Selected sources:

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