Abstract
In this paper, generalizing the notion of a path we define ak-area to be the setD={g(t):t ∈J} on thek-skeleton of a convex compact setK in a Hilbert space, whereg is a continuous injection map from thek-dimensional convex compact setJ to thek-skeleton ofK. We also define anE k-area onK, whereE k is ak-dimensional subspace, to be ak-area with the propertyπ(g(t))=t,t ∈π(K), whereπ is the orthogonal projection onE k. This definition generalizes the notion of an increasing path on the 1-skeleton ofK. The existence of such sets is studied whenK is a subset of a Euclidean space or of a Hilbert space. Finally some conjectures are quoted for the number of such sets in some special cases.
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Dalla, L. On a class of some special sets on thek-skeleton of a convex compact set. Israel J. Math. 68, 353–364 (1989). https://doi.org/10.1007/BF02764990
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DOI: https://doi.org/10.1007/BF02764990