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Let K ⊂ R d K \subset {{\mathbf {R}}^d} be a convex body, γ \gamma its center of mass. For Λ ⊂ R d \Lambda \subset {{\mathbf {R}}^d} a subspace of dimension d − k d - k , we establish the inequality \[ Vol d ( K ) ⩽ Vol d − k ( K | Λ ) Vol k ( ( K − γ ) ∩ Λ ⊥ ) {\operatorname {Vol} _d}(K) \leqslant {\operatorname {Vol} _{d - k}}(K|\Lambda ){\operatorname {Vol} _k}((K - \gamma ) \cap {\Lambda ^ \bot }) \] (where K | Λ K|\Lambda denotes orthogonal projection of K K onto Λ \Lambda ). Equality holds only if each k k -dimensional section of K K parallel to Λ ⊥ {\Lambda ^ \bot } is a translate of ( K − γ ) ∩ Λ ⊥ (K - \gamma ) \cap {\Lambda ^ \bot } .

The proximal point algorithm for solving 0 e T(x) with T maximal monotone is extended to mappings T satisfying the weaker property of maximal strict hypomonotonicity. The algorithm is applied to the minimization of a certain class of nondifferentiable nonconvex functions, the lower-C2 functions., whose subdifferentials are maximal strictly hypomonotone. For functions in this class, each step of the algorithm consists in minimizing a convex function.

An algorithm is described for finding a feasible point for a system of linear inequalities. If the solution set has nonempty interior, termination occurs after a finite number of iterations. The algorithm is a projection-type method, similar to the relaxation methods of Agmon, Motzkin, and Schoenberg. It differs from the previous methods in that it solves for a certain “dual” solution in addition to a primal solution.

ForT a maximal monotone operator on a Hilbert spaceH andA a closed subspace ofH, the “partial inverse”T A ofT with respect toA is introduced.T A is maximal monotone. The proximal point algorithm, as it applies toT A , results in a simple procedure, the “method of partial inverses”, for solving problems in which the object is to findx ∈ A andy ∈ A ⊥ such thaty ∈ T(x). This method specializes to give new algorithms for solving numerous optimization and equilibrium problems.

An algorithm previously introduced by the author for finding a feasible point of a system of linear inequalities is further investigated. For inconsistent systems, it is shown to generate a sequence converging at a linear rate to the set of least-squares solutions. The algorithm is a projection-type method, and is a manifestation of the proximal-point algorithm.

Fur a nonexpansive mappingq: R d →R d , we examine some properties of the setS of all possible limit points of sequences of the formx, q (x), q 2 (x), … We then characterize those piecewise isometric nonexpansive mappings having the property: for everyx there exists a finitek such thatq k (x)∈S.