On the Minkowski sum of two curves (Q1664054)

The aim of the present work is to answer a question posed by M. Laczkovich concerning the Minkowski sum of two curves in \({{\mathbb{R}}^{2}}\). More precisely, let \(A\subset \mathbb{R}^2\) be a continuous curve from \((0,0)\) to \((1,0)\), \(B\subset \mathbb{R}^2\) a continuous curve from \((0,0)\) to \((0,1)\) and \(S=A+B=\{a+b\mid a\in A, b\in B\}\). Then the author proves that the Lebesgue measure of the minimum possible area \(S\) is equal to \(1\).