On Volumes of Simplices in Intermediate Dimensions

A variant of the Falconer distance problem asks for fixed k 1 k ⩾ 1 and d k+1 d ⩾ k + 1 , how large does the Hausdorff dimension of a Borel set E⊂ Rᵈ E ⊂ R d need to be to guarantee that there exist x₀,… ,xₖ∈ E x 0 , … , x k ∈ E such that \,Vol\,ₖ₊₁^(x₀,… ,xₖ)(E) = \,Vol\,ₖ₊₁(x₀,… ,xₖ,xₖ₊₁): xₖ₊₁∈ E 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 \,Vol\,ₖ₊₁(x₀,… ,xₖ,xₖ₊₁) Vol k + 1 ( x 0 , … , x k , x k + 1 )