Laplace transforms and valuations

Legendre变换、Laplace变换与估值

The Legendre transform, the Laplace transform and valuations

本文研究Legendre变换、Laplace变换在不同函数类上的特征化，揭示其作为反变估值的唯一性和对偶变换的性质。

$\rm{SL}(n)$ covariant function-valued valuations

Classifications of $\rm{SL}(n)$ covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified characterizations of the Laplace transform on convex bodies, $L_p$ moment bodies, $L_p$ difference bodies, and polar $L_p$ moment bodies ($L_{-p}$ intersection bodies). Using the new classifications, we also establish some Euler-type relations.

凸函数上光滑赋值的 Paley-Wiener-Schwartz 定理

A Paley-Wiener-Schwartz Theorem for smooth valuations on convex functions

本文证明了凸函数上光滑赋值的 Paley-Wiener-Schwartz 定理，通过 Fourier-Laplace 变换刻画连续对偶上平移不变赋值，并以此推导平移群表示的光滑向量积分表示，最终实现闭和仿射不变子空间的完整分类及混合 Monge-Ampère 算子赋值的密度结果。

单位元不变估值在凸函数空间上

Unitarily invariant valuations on convex functions

研究复数空间中有限值凸函数的单位元不变估值，证明其可由特定子空间限制唯一确定。

$\operatorname{SL}(n)$ invariant valuations on super-coercive convex functions

All non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^n$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\operatorname{SL}(n)$ and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume and polar volume are characterized.

A Fourier type transform on translation invariant valuations on convex sets

Let $V$ be a finite dimensional real vector space. In the article we construct an isomorphism between the space of smooth translation invariant valuations on convex subsets of $V$ and the space of such valuations (twisted by densities) on the dual space $V^*$. This isomorphism commutes with the natural action of the full linear group on both spaces of valuations, maps product on the source (introduced in arXiv:math/0301148) to the convolution on the target (introduced in arXiv:math/0607449), and satisfies a Plancherel type inversion formula. As an application, a version of the hard Lefschetz t…