We present an analysis of the properties and characteristics of weakly
optimal entanglement witnesses, that is witnesses whose expectation value
vanishes on at least one product vector. Any weakly optimal entanglement
witness can be written as the form of Wwopt=σ−cσmaxI,
where cσmax is a non-negative number and I is the identity
matrix. We show the relation between the weakly optimal witness Wwopt and
the eigenvalues of the separable states σ. Further we give an
application of weakly optimal witnesses for constructing entanglement witnesses
in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et
al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric
properties.Comment: 13 pages, 2 figures, has been extensively redrafted and restructure