TOPICS
Search

Isotropic Constant

The isotropic constant of a convex body K subset R^n is the affine invariant quantity

L_K=((detSigma_K)/(vol_n(K)^2))^(1/(2n)),

where Sigma_K is the covariance matrix of the uniform distribution on K. Writing b_K for the geometric centroid,

Sigma_K=1/(vol_n(K))int_K(x-b_K)(x-b_K)^Tdx.

The definition is unchanged by an invertible affine transformation. For the interval [0,1], the variance is 1/12, so L_K=1/sqrt(12). For the hypercube [0,1]^n, independence gives the same isotropic constant.

The strong isotropic constant conjecture proposes that simplices maximize L_K in each dimension.

See also

Convex Body, Covariance Matrix, Differential Entropy, Strong Isotropic Constant Conjecture, Uniform Distribution

Explore with Wolfram|Alpha

References

Kipp, C. "Shadow Systems, Decomposability and Isotropic Constants." Disc. Comput. Geom. 76, 1391-1425, 2026. https://doi.org/10.1007/s00454-026-00829-7.

Cite this as:

Weisstein, Eric W. "Isotropic Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IsotropicConstant.html

Subject classifications