5 problems

Let $X\subseteq\mathbb{R}^n$ be an affine variety, let $K\subseteq\mathbb{R}^n$ be a generic linear subspace with $\dim K=k<n-d$, where $d=\dim X$, and let…

Let $K\subset\mathbb R^3$ be a convex body and let $p$ be a point in its interior. For $u\in\mathbb S^2$, let $\pi_u$ denote orthogonal projection in direction $u$. Point equichord…

Let $K,L\subset\mathbb R^3$ be convex bodies with $L\subset\operatorname{int}K$, and suppose that $L$ is strictly convex. For $u,w\in\mathbb S^2$ with $u\bot w$, let $\pi_u$ denote…

Fix $0\leq t/2\leq s\leq t\leq 1$, and let $K\subset\mathbb{R}^2$ be a Borel set with $\dim_{\mathrm{H}}K\geq t$. For $e\in S^1$, let $\pi_e:\mathbb{R}^2\to\mathbb{R}$ be the ortho…

Let $K\subset\mathbb{R}^{2}$ be a Borel set, and let $\pi_e\colon\mathbb{R}^{2}\to\mathbb{R}$ be the orthogonal projection $\pi_e(x)=x\cdot e$ for $e\in S^{1}$. Assume that…