A projection from X (Banach Space) onto V (its closed subspace):
- continuous linear operator from X onto V,
- Pv=v for all v in V.
The relative projection constant of V with respect to X:
- λ(V,X)=sup{||P||, P - projection from X onto V}.
A projection Q is minimal if it has the smallest possible norm:
- Q=λ(V,X).
Research directions:
- finding minimal projections,
- uniqueness of minimal projections,
- norm estimations.
The absolute projection constant of V:
- λ(V)=sup{λ(V,X), X contains V}=λ(V,L_infty).
Tough Question:
- characterize spaces V that maximize absolute projection constant over all spaces of a fixed dimension n.
For n=2 the answer is a space given by a regular hexagon:
- B.L. Chalmers and G.Lewicki, A proof of the Grunbaum conjecture, Studia Math. 200 (2010), no.2, 103-129.