Pinned nonempty interior and volumes of simplices

Abstract

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 $\Phi(x,y)$, 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 $\Delta_\Phi^y(E)=\{\Phi(x,y):x\in E\}$ 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.