固定非空内部与单纯形体积
Pinned nonempty interior and volumes of simplices
AI总结：
研究标量两点配置和单纯形体积的固定非空内部问题，利用光滑标量配置映射及相关计算，在一定维度条件下通过单频损失估计证明几乎每个固定点有连续固定密度和非空内部，还得到单纯形体积相关结论及应用。
AI中文摘要：
我们研究标量两点配置和单纯形体积的固定非空内部问题。对于\(E\subset\mathbb{R}^d\)（\(d\geq2\)）紧致且有光滑标量配置映射\(\Phi(x,y)\)，其相应局部广义拉东变换是非退化傅里叶积分算子，平滑阶为\((d - 1)/2\)。当\(\dim_{\mathcal H}(E)>(d + 1)/2\)时，可利用相关计算得到\(\Delta_\Phi^y(E)=\{\Phi(x,y):x\in E\}\)对几乎每个固定点\(y\)有正勒贝格测度。主要结果证明了在\(d\geq3\)且\(\dim_{\mathcal H}(E)>(d + 2)/2\)时，关于水平参数微分的相应单频损失估计产生连续固定密度，从而几乎每个固定点有非空内部。具体应用包括广义范数距离等。主要几何应用涉及单纯形体积，证明了\(\mathbb{R}^d\)中三角形面积的圆柱平均估计，在维度阈值\((d + 1)/2\)时双固定面积集有正测度，在\((d + 2)/2\)时有非空内部。投影定理将更高维单纯形体积问题简化为三角形面积问题。特别地，对于\(3\leq k \leq d\)，若\(\dim_{\mathcal H}(E)>(d + k - 1)/2\)，则对于每个规定基点\(x_0\)和每个规定第二个顶点\(y\in E\setminus\{x_0\}\)，由\(x_0,y\)和\(E\)的另外\(k - 1\)个点生成的\(k\)维体积集有非空内部。因此该结果在前两个顶点处是双重强固定的。
英文摘要：
We study pinned nonempty-interior problems for scalar two-point configurations and for volumes of simplices. For $E\subset\mathbb{R}^d$, $d\geq 2$, compact and a smooth scalar configuration map $Φ(x,y)$, whose corresponding localized generalized Radon transforms are nondegenerate Fourier integral operators of smoothing order $(d-1)/2$, we first note how a calculation due to Greenleaf, Iosevich and Taylor can be used to obtain positive Lebesgue measure of $Δ_Φ^y(E)=\{Φ(x,y):x\in E\}$ for almost every pin $y$ when $\dim_{\mathcal H}(E)>(d+1)/2$. Our first main result is to prove that the corresponding one-frequency-loss estimate for differentiation in the level parameter yields a continuous pinned density, and hence nonempty interior, for almost every pin when $d\geq3$ and $\dim_{\mathcal H}(E)>(d+2)/2$. Concrete applications include generalized norm distances, regular variable-coefficient and Riemannian distances, and dot products or nondegenerate bilinear forms on regular patches. Our principal geometric application concerns volumes of simplices. We prove a cylinder-averaging estimate for triangle areas in $\mathbb{R}^d$ and obtain positive measure for doubly pinned area sets at a dimensional threshold $(d+1)/2$ and nonempty interior at $(d+2)/2$. A projection theorem then reduces higher simplex-volume problems to triangle areas. In particular, for $3\leq k \leq d$, if $\dim_{\mathcal H}(E)>(d+k-1)/2$, then for every prescribed base point $x_0$ and every prescribed second vertex $y\in E\setminus\{x_0\}$, the set of $k$-dimensional volumes generated by $x_0,y$ and $k-1$ further points of $E$ has nonempty interior. Thus the result is doubly strongly pinned in its first two vertices.
