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Suppose y is a positive vector. Let p and x be two positive matrices with N rows, where ##p_j## and ##x_j## denotes the j:th row in these matrices, so that j = 1,…,N.

Does the following hold:

[tex]\inf_{k=1,...,N} [\sup_{l=1,...,N} [p_k(y-x_k)]] = \inf_{k=1,...,N} [p_k(y-x_k)][/tex]

where ##p_k(y-x_k)## is a scalar.

That is, given that I don’t take sup over the sub indices k but over the sub indices l which isn’t part of the expression ##p_k(y-x_k)## will I be able to skip the sup in the expression?

If this is true, then I should also be able to write

[tex]\sup_{k=1,...,N} [\inf_{l=1,...,N} [p_l(y-x_l)]] = \inf_{l=1,...,N} [p_l(y-x_l)][/tex]

Does the following hold:

[tex]\inf_{k=1,...,N} [\sup_{l=1,...,N} [p_k(y-x_k)]] = \inf_{k=1,...,N} [p_k(y-x_k)][/tex]

where ##p_k(y-x_k)## is a scalar.

That is, given that I don’t take sup over the sub indices k but over the sub indices l which isn’t part of the expression ##p_k(y-x_k)## will I be able to skip the sup in the expression?

If this is true, then I should also be able to write

[tex]\sup_{k=1,...,N} [\inf_{l=1,...,N} [p_l(y-x_l)]] = \inf_{l=1,...,N} [p_l(y-x_l)][/tex]

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