Take a point on the straight line B 1D so that PO 1, PO2 and PO3 are perpendicular to b 1, B 1c and B 1A in O 1, O2 and O3, respectively.

Then PO 1⊥ face A 1C 1,PO2⊥ face B 1C PO3⊥ face A 1B,

O 1, O2, O3 are O 1N ⊥ A 1D 1, O2M ⊥ CC 1, O3Q ⊥ AB respectively, and the vertical feet are m, n, q respectively.

Even PM, PN, PQ can be obtained from the three perpendicular theorems, pn ⊥ a1d1pm ⊥ cc1; PQ⊥AB，

Because all the faces in the cube are congruent, P0 1=PO2=PO3, ∴PM=PN=PQ, that is, the distance from P to the straight line of three sides AB, CC 1, A 1D 1. Are all equal, so there are infinite points to meet the conditions, so choose D.