On Volumes of Simplices in Intermediate Dimensions

Abstract

Abstract A variant of the Falconer distance problem asks for fixed $$k\geqslant 1$$ k ⩾ 1 and $$d\geqslant k+1$$ d ⩾ k + 1 , how large does the Hausdorff dimension of a Borel set $$E\subset \mathbb {R}^d$$ E ⊂ R d need to be to guarantee that there exist $$x_0,\ldots ,x_{k}\in E$$ x 0 , … , x k ∈ E such that $${{\,\textrm{Vol}\,}}_{k+1}^{(x_0,\ldots ,x_{k})}(E) = \lbrace {{\,\textrm{Vol}\,}}_{k+1}(x_0,\ldots ,x_{k},x_{k+1}): x_{k+1}\in E \rbrace $$ Vol k + 1 ( x 0 , … , x k ) ( E ) = { Vol k + 1 ( x 0 , … , x k , x k + 1 ) : x k + 1 ∈ E } has positive Lebesgue measure. Here $${{\,\textrm{Vol}\,}}_{k+1}(x_0,\ldots ,x_{k},x_{k+1})$$ Vol k + 1 ( x 0 , … , x k , x k + 1 )