Convex optimization with larger feasible set has more solutions, is this correct?

Question

Let $f:A\mapsto\mathbb{R}$ be a convex function with convex set $A$. Let $B\subset A$ be a convex subset. Denote \begin{equation} S_{A}:=\arg\min\limits_{x\in A}f(x),\quad S_{B}:=\arg\min\limits_{x\in B}f(x). \end{equation} I guess $|S_{A}|\geq |S_{B}|$, where $|\cdot|$ denote cardinality of a set. Is my proposition correct? Can anyone help me prove it or give some counter example?

Answer 1

Let $f(x_1, x_2) = x_1$.

With $B = \{(x_1, x_2) : x_1 \ge 0, x_2 \in [-1, 1]\}$, the minimizers of $f$ are $\{(0, x_2) : x_2 \in [-1, 1]\}$.

However for $A = \{(x_1, x_2) : |x_2| \le x_1 + 1\}$, $f$ has a unique minimizer at $(-1, 0)$.