Continuous, epi-translation invariant valuations on super-coercive convex functions decompose into homogeneous components of degrees 0 through n, and the degree-n valuations are exactly integrals of compactly supported functions of the gradient.
Dot product invariant valuations on Lip$(S^{n-1})$
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We provide an integral representation for continuous, rotation invariant and dot product invariant valuations defined on the space Lip$(S^{n-1})$ of Lipschitz continuous functions on the unit $n-$sphere.
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A homogeneous decomposition theorem for valuations on convex functions
Continuous, epi-translation invariant valuations on super-coercive convex functions decompose into homogeneous components of degrees 0 through n, and the degree-n valuations are exactly integrals of compactly supported functions of the gradient.