In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semi-algebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semi-algebraic convex sets and the dimension of the underlying space. We achieve our results by exploiting the algebraic structure of the basic semialgebraic convex sets.2010 Mathematics Subject Classification: Primary 41A25, 90C25; Secondary 41A50, 90C31

In this paper, we consider the structure of maximally monotone operators in Banach space whose domains have nonempty interior and we present new and explicit structure formulas for such operators. Along the way, we provide new proofs of the norm-toweak * closedness and of property (Q) for these operators (as recently proven by Voisei). Various applications and limiting examples are given.

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar's constraint qualification holds.In this paper, we prove the maximal monotonicity of A + B provided that A and B are maximal monotone operators such that dom A ∩ int dom B = ∅, A + N dom B is of type (FPV), and dom A ∩ dom B ⊆ dom B. The proof utilizes the Fitzpatrick function in an essential way.2010 Mathematics Subject Classification: Primary 47H05; Secondary 49N15, 52A41, 90C25

In this paper, we construct maximally monotone operators that are not of Gossez's dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC-functions will not always be a BC-function. This provides a negative answer to a challenging question posed by Stephen Simons. Among other consequences, we deduce that every Banach space which contains an isomorphic copy of the James space J or its dual J * , or c 0 or its dual ℓ 1 , admits a non type (D) operator.2010 Mathematics Subject Classification: Primary 47A06, 47H05; Secondary 47B65, 47N10, 90C25 imally monotone operator, monotone operator, multifunction, operator of type (BR), operator of type (D), operator of type (NI), partial inf-convolution, Schauder basis, set-valued operator, skew operator, space of type (D), uniqueness of extensions, subdifferential operator.

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds.In this paper, we prove the maximal monotonicity of A + ∂f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function satisfying dom A ∩ int dom ∂f = ∅. Moreover, A + ∂f is of type (FPV). The maximal monotonicity of A + ∂f when int dom A ∩ dom ∂f = ∅ follows from a result by Verona and Verona, which the present work complements.2010 Mathematics Subject Classification: Primary 47A06, 47H05; Secondary 47B65, 47N10, 90C25