  A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)}
representable if $S$ equals the projection of a set in higher dimensional space
which is describable by some Linear Matrix Inequality (LMI). The contributions
of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we
give an explicit construction of an SDP representation for
$\cv{\cup_{k=1}^mW_k}$. This provides a technique for building global SDP
representations from the local ones. (ii) For the SDP representability of a
compact convex semialgebraic set $S$, we prove sufficient condition: the
boundary $\bdS$ is positively curved, and necessary condition: $\bdS$ has
nonnegative curvature at smooth points and on nondegenerate corners. This
amounts to the strict versus nonstrict quasi-concavity of defining polynomials
on those points on $\bdS$ where they vanish. The gaps between them are $\bdS$
having positive versus nonnegative curvature and smooth versus nonsmooth
points. A sufficient condition bypassing the gaps is when some defining
polynomials of $S$ are sos-concave. (iii) For the SDP representability of the
convex hull of a compact nonconvex semialgebraic set $T$, we find that the
critical object is $\pt_cT$, the maximum subset of $\pt T$ contained in $\pt
\cv{T}$. We prove sufficient conditions for SDP representability: $\pt_cT$ is
positively curved, and necessary conditions: $\pt_cT$ has nonnegative curvature
at smooth points and on nondegenerate corners. The gaps between them are
similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition
is also discussed.
