Annali di Matematica Pura ed Applicata

, Volume 34, Issue 1, pp 105–112

# A curvature study of convex bodies in banach spaces

• E. R. Lorch
Article

## Summary

Some differential geometric properties of smooth convex bodies in aBanach space are studied. Two mappings, one of which is linear, from the space to its adjoint space receive special attention. if the curvature of the body is suitably bounded, these mappings are one to one and onto.

## Keywords

Geometric Property Convex Body Smooth Convex Curvature Study Adjoint Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Literatur

1. (1).
E. R. Lorch,Convexity and Normed Spaces, « Publications de l'Institut Mathématique de l'Académie Serbe des Sciences », IV, pp. 109–112, (1952)
2. (2).
See among others,S. Mazur,↦de|Über schrvache convergenz in der Raümen (L p), « Stud. Math. », Vol. 4 (1933), pp. 128–133;V. Smulian,Some geometrical properties of the unit sphere, « Rec. Math. », Vol. 48, NS 6, (1939), pp. 90–94;V. Smulian,Sur la dérivabilité de la nonme dans l'éspace de Banach, C. R. (Doklady), « Acad. Sc. URSS », (N. S.), 27 (1940). pp. 643–648;R. Fortet,Remarques sur les éspaces uniformément convexes, « Bull. Soc. Math. de France », Vol. 69 (1941), pp. 23–46. (Fortet introduces the notion of a uniformly regular norm; this condition seems to be equivalent to demanding the uniform existence of more than the first derivative);R. C. James,Orthogonality and linear functionals in normed linear spaces, « Trans. Amer. Math. Soc. », Vol. 61 (1947), pp. 264–292. (The derivative of principal interest toJames is that ofGateaux). The interested reader may wish to consult recent works ofE. S. Citlanadze andE. H. Rothe which have some points of common interest with the treatment of the present article.
3. (3).
P. C. Rosenbloom, Abstract 192, « Bull. Amer. Math. Soc. », Vol. 58, N. 2 (March. 1952).Google Scholar
4. (4).
E. R. Lorch,Differentiable inequalities and the theory of convex bodies, « Trans. Amer. Math. Soc. », Vol. 71 (1951), pp. 243–266. In particular, our present theorem 8 is the generalization toHilbert space of the fundamental homeomorphism needed to elaborate the foundation of theMinkowski theory of mixed volumes (see pp. 261–266).
5. (5).
S. Banach,Théorie des opérations linéaires, p. 23, théorème 4.Google Scholar
6. (6).
Loc. cit. in (4) See in paticular section II. The extension to an infinite number of dimensions presents no difficulties. In particular, if one considers the space spanned by two vectorsx andy, the results of theorem 1, p. 250, are directly applicable.
7. (7).
SeeBanach, loc. cit., p. 41, théorème 7.Google Scholar
8. (8).
Smulian gives conditions on the norm in order that the space be reflexive. SeeSmulian, loc. cit. in (2). The condition used byRosenbloom in order thatB be uniformly convex is not known to the author. SeeRosenbloom, loc. cit. in (3).Google Scholar