Hessian valuations

The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures

“…A functional Z : Conv(R n ; R) → R is dually epi-translation invariant if and only if Z(v+ℓ+γ) = Z(v) for every v ∈ Conv(R n ; R), every linear functional ℓ : R n → R and every γ ∈ R, or equivalently, if the map u → Z(u * ), defined on Conv sc (R n ), is epi-translation invariant. It was shown in [15] that…”

“…The following statement gathers properties of Monge-Ampère measures. Items (a) and (b) are due to Aleksandrov [1] (or see [21, Proposition 2.6 and Theorem A.31]) while the valuation property (c) was deduced by Alesker [4] from Błocki [9] (or see [15,Theorem 9.2]). Recall that for a sequence M k of Radon measures in R n , we say that M k converges weakly to a Radon measure…”

“…for every Borel function β : R n → [0, ∞). We remark that the measure Ψ j (u; •) can be defined for every u ∈ Conv sc (R n ) and is a marginal of a generalized Hessian measure (see [15]). Moreover, for…”

The Hadwiger theorem on convex functions, IV: The Klain approach

“…We recall the definition of two families of measures used in our results. They are both marginals of more general Hessian measures, see [8,12]. We remark that one of these families of Hessian measures was introduced by Trudinger and Wang [38,39].…”

“…It was shown in [12] that Z : X → R is a continuous valuation if and only if Z * : X * → R is a continuous valuation. Since u ∈ Conv sc (R n ) if and only if u * ∈ Conv(R n ; R), this allows us to transfer results between Conv sc (R n ) and Conv(R n ; R).…”