典型连续函数凸包的重分形性质
Multifractal properties of convex hulls of typical continuous functions
摘要：
研究了定义在$[0,1]^{d}$上的典型/一般连续函数的凸壳的奇异性（多重分形）谱。我们用${\mathbf E}_ { { φ} }^{h} $表示$ φ: [0,1]^d\to {\mathbb R}$的点态霍德指数等于$h$的点的集合。设$H_{f}$是$f$图的凸包，$H_{f}$顶部的凹函数用$ { { φ} }_{1,f}( { { \mathbf x } })=\max \{y:( { { \mathbf x } },y)\in H_{f} \}$表示，$ { { φ} }_{2,f}( { { \mathbf x } })=\min \{y:( { { \mathbf x } },y)\in H_{f} \}$表示$H_{f}$底部的凸函数.我们表明，有一个密集的$G_δ$子集$ { { \cal G } } { \subset } {C[0,1]^d}$，使$f\in { { \cal G } }$满足以下属性。对于$i=1,2$，函数$ { { { φ} }_ {i,f}}$和$f$仅在零Hausdorff维数的集合上重合，函数$ { { { φ} }_ {i,f}}$在$(0,1)^{d}$上连续可微，如果$h\in(0,+ { \infty }) { \setminus } \{1 \}$，则${\mathbf E}_{ { { φ} }_{i,f}}^{0} $等于$ {[0,1]^d}$、$\dim_{H}{\mathbf E}_{ { { φ} }_{i,f}}^{1}=d-1 $、$\dim_{H}{\mathbf E}_{ { { φ} }_{i,f}}^{+ { \infty }}=d $和${\mathbf E}_{ { { φ} }_{i,f}}^{h}= { \emptyset }$的边界。
英文摘要：
We study the singularity (multifractal) spectrum of the convex hull of the typical/generic continuous functions defined on $[0,1]^{d}$. We denote by ${\mathbf E}_ { { φ} }^{h} $ the set of points at which $ φ: [0,1]^d\to {\mathbb R}$ has a pointwise Hölder exponent equal to $h$. Let $H_{f}$ be the convex hull of the graph of $f$, the concave function on the top of $H_{f}$ is denoted by $ { { φ} }_{1,f}( { { \mathbf x } })=\max \{y:( { { \mathbf x } },y)\in H_{f} \}$ and $ { { φ} }_{2,f}( { { \mathbf x } })=\min \{y:( { { \mathbf x } },y)\in H_{f} \}$ denotes the convex function on the bottom of $H_{f}$. We show that there is a dense $G_δ$ subset $ { { \cal G } } { \subset } {C[0,1]^d}$ such that for $f\in { { \cal G } }$ the following properties are satisfied. For $i=1,2$ the functions $ { { { φ} }_ {i,f}}$ and $f$ coincide only on a set of zero Hausdorff dimension, the functions $ { { { φ} }_ {i,f}}$ are continuously differentiable on $(0,1)^{d}$, ${\mathbf E}_{ { { φ} }_{i,f}}^{0} $ equals the boundary of $ {[0,1]^d}$, $\dim_{H}{\mathbf E}_{ { { φ} }_{i,f}}^{1}=d-1 $, $\dim_{H}{\mathbf E}_{ { { φ} }_{i,f}}^{+ { \infty }}=d $ and ${\mathbf E}_{ { { φ} }_{i,f}}^{h}= { \emptyset }$ if $h\in(0,+ { \infty }) { \setminus } \{1 \}$.
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