Let $f:2^V\to \mathbb{R}$ be a symmetric submodular function. Let $T\subset V$, consider the function $f_T:2^T\to \mathbb{R}$ defined as follows: $$f_T(X) = \min \{f(Y) | X\subset Y,T\backslash X\subset V\backslash Y \}$$
Is $f_T$ submodular? If so, do we need $f$ to be symmetric? It be best if there is a known reference.
I can prove each condition considered independently is submodular when $f$ is submodular. say $g_T,\overline{g_T}:2^T\to \mathbb{R}$ defined as $$g_T(X) = \min \{f(Y) | X\subset Y\}$$ $$\overline{g_T}(X) = \min \{f(Y) | T\backslash X\subset V\backslash Y\}$$
Lemma 1: Let $$f_*(X) = \min \{ f(Y) : Y\subset X\}$$ and $$f^*(X) = \min \{ f(Y) : X \subset Y\}$$. If $f$ is submodular then $f_*$ and $f^*$ are submodular.
This can be proven easily from definition, and one can find reference for $f_*$ as proposition 2.1 in Submodular functions and convexity.
If $f:2^V \to \mathbb{R}$, then define $f':2^V \to \mathbb{R}$ as $f'(X)=f(V\backslash X)$.
Lemma 2: $f$ is submodular if and only if $f'$ is submodular.
$g_T$ is submodular because $g_T$ is just $f^*$ restricted on $T$.
$\overline{g_T}$ is submodular \begin{align*} (\overline{g_T})'(X) &= \overline{g_T}(T\backslash X)\\ &=\min \{f(Y) | T\backslash (T\backslash X)\subset V\backslash Y\}\\ &=\min \{f(Y) | X\subset V\backslash Y\}\\ &=\min \{f(Y) | Y\subset V\backslash X\}\\ &= (f_*)'(X) \\ \end{align*}
So $(\overline{g_T})'$ is $(f_*)'$ restricted to $T$, thus $\overline{g_T}$ is submodular.
