  A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $
halfplanes with common boundary $ l $, a line, where the vertices are on $ l $
and the edges cannot cross $ l $. The $ k $-page book crossing number of the
graph $ G $, denoted by $ \nu_k(G) $, is the minimum number of edge-crossings
over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph
on $n$ vertices. We improve the lower bounds on $ \nu_k(K_n) $ for all $ k\geq
14 $ and determine $ \nu_k(K_n) $ whenever $ 2 < n/k \leq 3 $. Our proofs rely
on bounding the number of edges in convex graphs with small local crossing
numbers. In particular, we determine the maximum number of edges that a convex
graph with local crossing number at most $ \ell $ can have for $ \ell\leq 4 $.
