  Logconcave functions represent the current frontier of efficient algorithms
for sampling, optimization and integration in R^n. Efficient sampling
algorithms to sample according to a probability density (to which the other two
problems can be reduced) relies on good isoperimetry which is known to hold for
arbitrary logconcave densities. In this paper, we extend this frontier in two
ways: first, we characterize convexity-like conditions that imply good
isoperimetry, i.e., what condition on function values along every line
guarantees good isoperimetry? The answer turns out to be the set of
(1/(n-1))-harmonic concave functions in R^n; we also prove that this is the
best possible characterization along every line, of functions having good
isoperimetry. Next, we give the first efficient algorithm for sampling
according to such functions with complexity depending on a smoothness
parameter. Further, noting that the multivariate Cauchy density is an important
distribution in this class, we exploit certain properties of the Cauchy density
to give an efficient sampling algorithm based on random walks with a mixing
time that matches the current best bounds known for sampling logconcave
functions.
