TY - UNPB
T1 - Lipschitz constant log{n} almost surely suffices for mapping n grid points onto a cube
AU - Dymond, Michael
N1 - 18 pages
PY - 2020/10/28
Y1 - 2020/10/28
N2 - Kaluža, Kopecká and the author have shown that the best Lipschitz constant for mappings taking a given nd-element set in the integer lattice ℤd, with n∈ℕ, surjectively to the regular n times n grid {1,…,n}d may be arbitrarily large. However, there remain no known, non-trivial asymptotic bounds, either from above or below, on how this best Lipschitz constant grows with n. We approach this problem from a probabilistic point of view. More precisely, we consider the random configuration of nd points inside a given finite lattice and establish almost sure, asymptotic upper bounds of order log n on the best Lipschitz constant of mappings taking this set surjectively to the regular n times n grid {1,…,n}d.
AB - Kaluža, Kopecká and the author have shown that the best Lipschitz constant for mappings taking a given nd-element set in the integer lattice ℤd, with n∈ℕ, surjectively to the regular n times n grid {1,…,n}d may be arbitrarily large. However, there remain no known, non-trivial asymptotic bounds, either from above or below, on how this best Lipschitz constant grows with n. We approach this problem from a probabilistic point of view. More precisely, we consider the random configuration of nd points inside a given finite lattice and establish almost sure, asymptotic upper bounds of order log n on the best Lipschitz constant of mappings taking this set surjectively to the regular n times n grid {1,…,n}d.
KW - math.FA
KW - 51F99, 51M05, 52C99, 26B10
U2 - 10.48550/arXiv.2010.15073
DO - 10.48550/arXiv.2010.15073
M3 - Preprint
BT - Lipschitz constant log{n} almost surely suffices for mapping n grid points onto a cube
PB - arXiv
ER -