Proximinal faces of some Banach spaces

In this paper, we discover more proximinal closed convex subsets of unit spheres of some non-reflexive Banach spaces. Let B be a Banach space. For an element, u belongs to the unit sphere S_B , and for a norm attaining functional μ on B , the sets Q ( u ) and 𝒥_B(μ ) are faces of S_B. We consider B to be C_0(X) , the space of all real-valued continuous functions vanishing at infinity on a locally compact Hausdorff space X , and L_1(K,μ ), the space of all μ -measurable functions on a compact Hausdorff space K . In each space, we prove that the face Q ( u ) is strongly proximinal in B . For B=A(K), the space of all affine continuous functions on a Choquet simplex K , we prove sufficient conditions for the faces Q ( u ) and 𝒥_B(μ ) to be proximinal in B . If Y is the space of all self-adjoint compact operators on l_2 , then Y is a closed subspace of K(l_2) , the space of all compact operators on l_2 . We characterize the faces Q ( u ) and 𝒥_Y(μ ) of Y and discuss their proximinality properties.