Analytic solutions of convolution equations on convex sets with a mixed structure. I

On regularization in superreflexive Banach spaces by infimal convolution formulas

We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with $α$-Hölder derivatives (for some $0<α\leq 1$). The smooth approximation is given by means of an explicit formula enjoying good properties from the minimization point of view. For instance, for any function $f$ which is bounded below and uniformly continuous on bounded sets this formula gives a sequence of $Δ$-convex ${\Cal{C}}^{1,α}$ functions converging uniformly on bounded sets to $f$ and preserving the infimum and the set of minimizers of $f$. The techniques we d…

On perturbation of a surjective convolution operator

Let $μ\in {\cal E}'({\mathbb R}^n)$ be a compactly supported distribution such that its support is a convex set with non-empty interior. Let $X_2$ be a convex domain in ${\mathbb R}^n$, $X_1 = X_2 + supp \ μ$. Assuming that a convolution operator $A: {\cal E}(X_1) \to {\cal E}(X_2)$ acting by the rule $(Af)(x) = (μ* f)(x)$ is surjective we provide a condition on a linear continuous operator $B: {\cal E}(X_1) \to {\cal E}(X_2)$ that guarantees surjectivity of the operator $A+B$.

Convolutions and multiplier transformations of convex bodies

Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Min…

凸函数的加法运动公式

Additive kinematic formulas for convex functions

本文通过Hadwiger定理和Kubota型公式，证明了凸函数的加法运动公式，并给出了功能内在体积表示等价性的新解释及混合面积测度的积分几何公式。

Functional Brunn-Minkowski Inequalities Induced by Polarity

We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geom…

加权布兰姆-闵科夫斯基理论II：混合测度的不等式及其应用

Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications

本文提出混合测度的不等式及其实用性，揭示了测度的超模性与Lebesgue测度的关系。

Soprunov和Zvavitch关于混合体积的Bezout不等式中的极值子

Extemizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes

本文研究了凸体满足Soprunov和Zvavitch提出的混合体积Bezout不等式的边界结构必要条件，并给出了三维情形的新解。

Bezout Inequality for Mixed volumes

In this paper we consider the following analog of Bezout inequality for mixed volumes: $$V(P_1,\dots,P_r,Δ^{n-r})V_n(Δ)^{r-1}\leq \prod_{i=1}^r V(P_i,Δ^{n-1})\ \text{ for }2\leq r\leq n.$$ We show that the above inequality is true when $Δ$ is an $n$-dimensional simplex and $P_1, \dots, P_r$ are convex bodies in $\mathbb{R}^n$. We conjecture that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $Δ$ must be an $n$-dimensional simplex. We prove that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $Δ$ must be indecomposable (i.e. cannot be…

Kubota型公式与混合测度的支持

Kubota-type formulas and supports of mixed measures

本文提出了一类新的Kubota型公式，用于描述适应旋转的混合面积测度，并通过刻画支持结构验证了Schneider的猜想。

$C_b$ 上混合拓扑中单调泛函的下半连续性

Lower semicontinuity of monotone functionals in the mixed topology on $C_b$

本文刻画 $C_b$ 上单调泛函的从下连续性与混合拓扑下半连续性的等价关系，并由此推导凸单调映射的可数可加测度对偶表示、Choquet 积分及等度连续性结论。