Every Banach space with a non-shrinking (unconditional) basis (Xi ) can be renormed so that the biorthogonal sequence  has a much smaller (unconditional) basis constant than (xi ). On the other hand, if the unconditional constant of
 has a much smaller (unconditional) basis constant than (xi ). On the other hand, if the unconditional constant of  is C < 2 then the unconditional constant of (xi ) is at most C/(2—C). This estimate is sharp.
 is C < 2 then the unconditional constant of (xi ) is at most C/(2—C). This estimate is sharp.