Abstract
Consider the projections of a finite set \(A\subset {\mathbb R}^n\) onto the coordinate hyperplanes; how small can the sum of the sizes of these projections be, given the size of A? In a different form, this problem has been studied earlier in the context of edge-isoperimetric inequalities on graphs, and it can be derived from the known results that there is a linear order on the set of n-tuples with non-negative integer coordinates, such that the sum in question is minimised for the initial segments with respect to this order. We present a new, self-contained and constructive proof, enabling us to obtain a stability result and establish algebraic properties of the smallest possible projection sum. We also solve the problem of minimising the sum of the sizes of the one-dimensional projections.
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Acknowledgements
We thank Ben Barber and Larry Harper who kindly helped us to rectify our ignorance about the edge-isoperimetric inequalities, and the anonymous referee for the helpful feedback.
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Lev, V.F., Rudnev, M. Minimising the Sum of Projections of a Finite Set. Discrete Comput Geom 60, 493–511 (2018). https://doi.org/10.1007/s00454-018-9975-2
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DOI: https://doi.org/10.1007/s00454-018-9975-2