Prescribing Tangent Hyperplanes to C ^1,1 and C ^{1, ω} Convex Hypersurfaces in Hilbert and Superreflexive Banach Spaces

Let

denote

\mathbb{R}^n R n

or, more generally, a Hilbert space. Given an arbitrary subset

C C

of

X X

and a collection

\mathcal{H} H

of affine hyperplanes of

X X

such that every

H\in\mathcal{H} H ∈ H

passes through some point

x_{H}\in C x H ∈ C

, and

C=\{x_H: H\in\mathcal{H}\} C = { x H : H ∈ H }

, what conditions are necessary and sufficient for the existence of a

C^{1,1} C 1 , 1

convex hypersurface

S S

in

X X

such that

H H

is tangent to

S S

at

x_H x H

for every

H\in\mathcal{H} H ∈ H

? In this paper we give an answer to this question. We also provide solutions to similar problems for convex hypersurfaces of class

C^{1, \omega} C 1 , ω

in Hilbert spaces, and for convex hypersurfaces of class

C^{1, \alpha} C 1 , α

in superreflexive Banach spaces having equivalent norms with moduli of smoothness of power type

1+\alpha 1 + α

,

\alpha\in (0, 1] α ∈ ( 0 , 1 ]

.