In 1946 Ambrose returned to UCL as an Assistant Lecturer and began a most fruitful collaboration with Harold Davenport on the Geometry of Numbers. In 1949 he went to the Institute of Advanced Study in Princeton as a Commonwealth Fund Fellow and teamed up with Dvoretsky to produce their famous result on absolute and unconditional convergence. Leaving a Readership at UCL, in 1954 Ambrose went to Birmingham as Mason Professor of Pure Mathematics. In collaboration with Geoffrey Shephard and James Taylor during that period his interest in convex geometry and Hausdorff Measure Theory widened. In particular, with Geoffrey Shephard, he produced sharp bounds for the volume of a difference body, a problem which had been open for 30 years.
A Junior Berwick Prize followed in 1957 and when Davenport moved to Cambridge in 1958, Ambrose returned to UCL as Astor Professor of Pure Mathematics. He was deemed to be too young to be the sole Head of Department and so for the next 28 years he was Joint Head, firstly with W.R. Dean and later with Keith Stewartson. Throughout this period he was the Principal Editor of Mathematika a journal that Harold Davenport had founded with the purpose of fast publication of results. In 1959 he was elected to the Royal Society and in 1961 spent a year in Canada where, in particular, working with Maurice Sion he developed an interest in analytic sets and put the final touches to his influential book Packing and Covering.
During the 1960s, Ambrose concentrated mainly on Hausdorff Measure Theory. This culminated in a wonderful example (with Roy Davies) of a compact metric space of infinite Hausdorff measure which has no subsets of finite positive measure. His book Hausdorff Measures is a standard text. From 1970-2, following the untimely death of Sir Edward Collingwood, he took over the Presidency of the LMS and in 1977 received the Society's highest honour, the De Morgan Medal. He was also Vice-President from 1958-59 and 1972-74.
During the 1970s his interests switched back to convex sets with spectacular success. This second period produced the famous work on the measure of the directions of line segments on the boundary of a convex body and a 12-dimensional counterexample to the Busemann-Petty problem. Retiring in 1986, he continued to work mainly on analytic sets, in the context of functional analysis, with John Jayne and Isaac Namioka.
Ambrose was a passionate supporter of the LMS and attended every London Meeting until his health began to fail. He was also passionate about research. I recall being summoned to his home to discuss research while he lay in bed recovering from pneumonia. His wide interests and depth of thought meant that most visitors to the UCL Mathematics Department ended up collaborating on a joint project with him.
He was married in 1952 to Joan North, a writer of children's books, who died in 1999. They had two daughters, Jane and Petra. Ambrose was also very proud of the achieve-ments of his nephew L.C.G. (Chris) Rogers.
David Larman University College London