ENCYCLOPEDIA OF DISTANCES

Nice distances (as a
perspective) images: **
1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15**

Encyclopedia of Distances (4th edition) - CORRECTIONS, ADDITIONS AND UPDATES:

\item{\index{\bf Minkowski length of lattice polytope}}

Let $P$ be a convex $d$-dimensional {\em lattice polytope} in $\mathbb {R}^d$, i.e., a convex hull of finitely many points in the integer lattice $\mathbb{Z}^d \subset \mathbb{R}^d$. The {\em lattice diameter} $l((P)$ is defined as one less than the largest number of collinear lattice points in $P$.

For any $1\le n\le d$, the {\em $n$-th Minkowski length} $L_n(P)$ is defined (Soprunov--Soprunova, 2009) as the largest number of lattice polytopes of positive dimension whose Minkowski sum is contained in $P$, i.e., the largest number of lattice segments whose Minkowski sum is at most $n$-dimensional and is contained in $P$; so, $L_1(P)=l(P)$. The {\bf Minkowski length} of $P$ is $L_d(P)$.

The largest synchronized movement of biomass on Earth is diel vertical migration of zooplankton in the ocean and lakes.

$50-300$ km: unexplored slice of atmosphere, where the air is too thin to support research balloons, but it is too thick for satellites to survive the drag forces for more than a few months.

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FURTHER COMMENTS should be sent to Michel Deza at this address: Michel.Deza@ens.fr