On Variations of Curves and Compact Sets

We consider various versions of the concepts of multidimensional variation and hypervariation for sets and curves (the hypervariation of a finite sequence of points is defined as the maxixum variation of its subsequences). We investigate their properties and establish relations between them. One of the main results of the paper says that the minimum dilation of curves that fill a connected compact set $K$ is equal to the hypervariation of $K$. We also establish a relationship between the variation and dilation for polyfractal curves. We calculate the two-dimensional (hyper)variations of some plane compact sets. In particular, we give a negative answer to the question of whether the two-dimensional variations of a convex compact set and of its boundary coincide. At the same time, a number of calculations testify to the conjecture that the two-dimensional variations of the unit square and of its boundary are equal, i.e., that the Kakutani–Gomori “two-dimensional length” of the unit square is $3$. The $d$-dimensional hypervariations of three- and four-dimensional Boolean cubes $\{0,1\}^n$ considered with the Hamming metric are also calculated.