For convex bodies, if an anisotropic area measure of order k is proportional to the top-order area measure, the body must be homothetic to a k-tangential body of the gauge body.
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Anisotropic area measures of convex bodies
For convex bodies, if an anisotropic area measure of order k is proportional to the top-order area measure, the body must be homothetic to a k-tangential body of the gauge body.