permutahedral

— This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every (1#43;#92;epsilon)-approximation of the \v{C}ech filtration has to contain n#123;#92;Omega(#92;log#92;logn)#125; features, provided that #92;epsilonlt;#92;frac#123;1#125;#123;#92;log#123;1#43;c#125;n#125; for c#92;in(0,1)..