Doping-induced evolutions of PbWO4 mesocrystals and their optical properties

Y-doped PbWO4 mesocrystals with controllable morphologies, from convex quadrangle to concave erythrocyte, are obtained by simply increasing the Y3+ doping concentrations.

Metric spaces which cannot be isometrically embedded in Hilbert space

Let A1A2A3A4, be a planar convex quadrangle with diagonals A1A3 and A2A4. Is there a quadrangle B1B2B3B4 in Euclidean space such that A1A3 < B1B3, A2A4 < B2B4 but AiAj > BiBj for other edges?The answer is “no”. It seems to be obvious but the proof is more difficult. In this paper we shall solve similar more complicated problems by using a higher dimensional geometric inequality which is a generalisation of the well-known Pedoe inequality (Proc. Cambridge Philos. Soc.38 (1942), 397–398) and an interesting result by L.M. Blumenthal and B.E. Gillam (Amer. Math. Monthly50 (1943), 181–185).