Magnitude's conjecture for intrinsic volumes of convex Euclidean sets

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Let AA be a compact convex subspace of ℓ2N\ell_2^N. For 0≤i≤N0\leq i\leq N, let Vi(A)V_i(A) denote its ii-th intrinsic volume, and let ωi\omega_i be the volume of the Euclidean unit ball in Ri\mathbb{R}^i. The magnitude–intrinsic-volume conjecture.

∣A∣=∑i=0N1i!ωiVi(A).|A|=\sum_{i=0}^N\frac{1}{i!\omega_i}V_i(A).

This conjecture predicts that the magnitude of every compact convex Euclidean set is determined by all of its intrinsic volumes, generalizing the exact formulas known in low-dimensional and special cases. Its general validity remains open.

References

Primary source

Tom Leinster, “The magnitude of metric spaces”, arXiv:1012.5857 (2011).

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