Juan Jorge Schäffer

On the geometry of spheres in L-spaces

It is shown that any two points on the surface of the unit ball ofL1(μ), where the measureμ is non-atomic, may be joined in the surface by a curve whose length is equal to the straight-line distance between its endpoints. This property is contrasted with the metric properties of the unit sphere in other L-spaces.

The structure of translation-invariant memories

under “Carathkodory conditions” that are, at least apparently, more general than those usually encountered in the literature. The values of u and r are in a Banach space E, r is locally integrable; and the “memory” M transforms continuous functions linearly into locally integrable functions, and “is ignorant of the future.” The domain of the independent variable t is R = ]--co, co[ or [0, co[; and certain further conditions are imposed on Ad. For instance, Ad is a “short memory” if the values of Mu on [a, b] depend on u only through its values on [a 1, b]; the choice of 1 for the maximum scope of the memory is made for the purposes of normalization. It is natural to enquire about the structure of the memory M when the equation (1.1) is autonomous; in that case it is natural to let the domain of t be R, and we shall do so. It is then obvious that the homogeneous equation zi + Mu = 0 is autonomous if and only if M commutes with all translations; for this reason we speak of translation-invariant memories. We shall restrict our attention in this paper to translation-invariant short memories. The structure of such a memory M is well known when E is finitedimensional and M is a continuous operator on the FrCchet space of continuous functions with values in E (the “continuous case”): ill is then expressed as a Stieltj& convolution with an operator-valued measure (cf. Theorem 8.7; see [6]). W e aim at some insight into the structure of M under more general “CarathCodory conditions” that will still ensure existence, uniqueness, and reasonable growth properties of the solutions of the initial-

Symmetric curves, hexagons, and the girth of spheres in dimension 3

It is proved that every centrally symmetric simple closed curve on the boundary of a centrally symmetric convex body in a three-dimensional linear space possesses an inscribed concentric affinely regular hexagon. This result is used to settle affirmatively a conjecture in [2] about the metric structure of the unit spheres of three-dimensional normed space.

Order-isomorphisms between cones of continuous functions

SummaryThis paper studies the structure of order-isomorphisms between the cones of strictly-positive-valued elements of subspaces of two spaces of continuous functions; a complete description is obtained in some cases. Conditions are also found that ensure that such an order-isomorphism is linear provided it is homogeneous along at least one ray.

Uniqueness without continuous dependence in infinite dimension

It is well known that the fundamental existence theorem for the initialvalue problem for an ordinary differential equation with continuous righthand side (Peano’s theorem) cannot be validly generalized to equations in an infinite-dimensional Banach space (Dieudonne [Z]; see [ 1; Example 2.11); there is, indeed, no infinite-dimensional Banach space to which it can be generalized (Godunov [3]). What about the analogous question concerning the assertion, valid for finite dimension, that “Uniqueness implies continuous dependence on the initial state?” (Uniqueness here means the existence of at most one noncontinuable solution for each initial time and state.) Since we have not found this question addressed in the literature, we have devised an example of an autonomous equation, in an infinite-dimensional Banach space, for which the implication fails in a strong way. We denote by N the set of all nonzero natural numbers. Then R” denotes the linear space of all sequences of real numbers, and I” denotes the Banach space of all bounded sequences, with the supremum norm. For each m E N we denote by e, the unit vector in I” defined by (e,), : = 6,, for all 12 E N. We denote the identity function and the square-root function from [0, co [ to [0, cc [ by z and ,/‘, respectively. Functional composition is indicated by 0.

More distant than the antipodes

In a normed space X, 6 denotes the inner metric on the surface d£ of the unit ball. We consider M(X) = sup{6(-p,p): p e d£} , D(X) = sup{6(p,q): p,q e 5E) . Exploding the conjecture that M(X) = D(X) for all X (previously verified in several cases), it is shown that M(X) = 2 , D(X) = 3 for X = CO((O,1]). *This work was supported in part by NSF Grant GP-19126 MORE DISTANT THAN THE ANTIPODES J. J. Schaffer*

Dichotomies for linear differential equations with delays : the Caratheodory case

SummaryThis work is concerned with differential equations with delays, on [0, ∞[, of the form

Exponential dichotomies for linear differential equations with delays: Periodic and autonomous equations

\dot u