  We consider assortment optimization over a continuous spectrum of products
represented by the unit interval, where the seller's problem consists of
determining the optimal subset of products to offer to potential customers. To
describe the relation between assortment and customer choice, we propose a
probabilistic choice model that forms the continuous counterpart of the widely
studied discrete multinomial logit model. We consider the seller's problem
under incomplete information, propose a stochastic-approximation type of
policy, and show that its regret -- its performance loss compared to the
optimal policy -- is only logarithmic in the time horizon. We complement this
result by showing a matching lower bound on the regret of any policy, implying
that our policy is asymptotically optimal. We then show that adding a capacity
constraint significantly changes the structure of the problem: we construct a
policy and show that its regret after $T$ time periods is bounded above by a
constant times $T^{2/3}$ (up to a logarithmic term); in addition, we show that
the regret of any policy is bounded from below by a positive constant times
$T^{2/3}$, so that also in the capacitated case we obtain asymptotic
optimality. Numerical illustrations show that our policies outperform or are on
par with alternatives.
