Q: smallest n-cube containing k points in euclidean n-space?
Jobst Heitzig
heitzig-j at web.de
Tue Jun 3 12:57:57 PDT 2003
Hello everybody!
Some problems:
Q1: Given k points in Euclidean n-space,
what is the smallest n-cube containing all of them?
Probably equivalent question:
Q2: Given
X = some k-dimensional metric space
Y = n-th power of unit interval, with euclidean metric
Can X be embedded isometrically into Y?
And finally my main problem:
Q3: Given
M = some k-dimensional correlation matrix
(i.e. positive semi-definite, all entries in [-1,1],
unit diagonal)
Y = n-th power of interval [-b,b] for some positive b
Are there k vectors x_1,...,x_k in Y such that
M_ij is the scalar product of x_i and x_j for all i,j?
(In other words: When can we realize a correlation matrix of k variables
with n observations, where the variables satisfy some given bounds?)
Jobst Heitzig,
Statistisches Bundesamt Deutschland
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