Hilbert's third problem

Target entity: Hilbert's third problem
Context triple: [Dehn invariant, relatedTo, Hilbert's third problem]

• A. Hilbert’s twenty-third problem
  Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
• B. Dehn invariant
  The Dehn invariant is a mathematical quantity in geometry that helps determine whether two polyhedra are scissors-congruent, playing a key role in the solution of Hilbert’s third problem.
• C. Minkowski’s theorem on convex sets
  Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
• D. Banach–Tarski paradox
  The Banach–Tarski paradox is a theorem in set-theoretic geometry stating that a solid ball in 3‑dimensional space can be decomposed into finitely many non-measurable pieces and reassembled into two identical copies of the original ball, highlighting counterintuitive consequences of the axiom of choice.
• E. Hilbert problems
  The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Hilbert's third problem
Target entity description: Hilbert's third problem is one of David Hilbert’s famous list of 23 problems, asking whether every polyhedron of a given volume is equidecomposable with any other of the same volume, a question that led to the development of the Dehn invariant and the discovery of counterexamples.

• A. Hilbert’s twenty-third problem
  Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
• B. Dehn invariant
  The Dehn invariant is a mathematical quantity in geometry that helps determine whether two polyhedra are scissors-congruent, playing a key role in the solution of Hilbert’s third problem.
• C. Minkowski’s theorem on convex sets
  Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
• D. Banach–Tarski paradox
  The Banach–Tarski paradox is a theorem in set-theoretic geometry stating that a solid ball in 3‑dimensional space can be decomposed into finitely many non-measurable pieces and reassembled into two identical copies of the original ball, highlighting counterintuitive consequences of the axiom of choice.
• E. Hilbert problems
  The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
• F. None of above. chosen