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We generalize a result by Cook, Magyar, and Pramanik [3] on three-termarithmetic progressions in subsets of $\mathbb{R}^d$ to corners in subsets of$\mathbb{R}^d\times\mathbb{R}^d$. More precisely, if $1<p<\infty$, $p\neq 2$,and $d$ is large enough, we show that an arbitrary measurable set$A\subseteq\mathbb{R}^d\times\mathbb{R}^d$ of positive upper Banach densitycontains corners $(x,y)$, $(x+s,y)$, $(x,y+s)$ such that the $\ell^p$-norm ofthe side $s$ attains all sufficiently large real values. Even though we closelyfollow the basic steps from [3], the proof diverges at the part relying onharmonic analysis. We need to apply a higher-dimensional variant of amultilinear estimate from [5], which we establish using the techniques from [5]and [6].

On side lengths of corners in positive density subsets of the Euclidean space