Let
K
K
be the basic closed semi-algebraic set in
R
n
\mathbb {R}^n
defined by some finite set of polynomials
S
S
and
T
T
, the preordering generated by
S
S
. For
K
K
compact,
f
f
a polynomial in
n
n
variables nonnegative on
K
K
and real
ϵ
>
0
\epsilon >0
, we have that
f
+
ϵ
∈
T
f+\epsilon \in T
[15]. In particular, the
K
K
-Moment Problem has a positive solution. In the present paper, we study the problem when
K
K
is not compact. For
n
=
1
n=1
, we show that the
K
K
-Moment Problem has a positive solution if and only if
S
S
is the natural description of
K
K
(see Section 1). For
n
≥
2
n\ge 2
, we show that the
K
K
-Moment Problem fails if
K
K
contains a cone of dimension 2. On the other hand, we show that if
K
K
is a cylinder with compact base, then the following property holds:
\[
(
‡
)
∀
f
∈
R
[
X
]
,
f
≥
0
on
K
⇒
∃
q
∈
T
such that
∀
real
ϵ
>
0
,
f
+
ϵ
q
∈
T
.
(\ddagger )\quad \quad \forall f\in \mathbb {R}[X], f\ge 0 \text { on } K\Rightarrow \exists q\in T \text { such that }\forall \text { real } \epsilon >0, f+\epsilon q\in T.\quad
\]
This property is strictly weaker than the one given in [15], but in turn it implies a positive solution to the
K
K
-Moment Problem. Using results of [9], we provide many (noncompact) examples in hypersurfaces for which (
‡
\ddagger
) holds. Finally, we provide a list of 8 open problems.