Open Convex Subsets of Dense Spaces

Question

So I asked this question yesterday, Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric, and it made my start wondering the following...

Suppose the following:

• $Y$ is a topological vector space
• X is a linear subspace of Y endowed with the subspace topology
• X is dense in Y
• U is open and convex in X.
• V is an open set in Y s.t. $U=V\cap X$
• $Conv(V)$ is the convex hull of V

Question: Is it necessary that $U=Conv(V)\cap X=V\cap X$?