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Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\top x$, and a corresponding distance $d_C$ on $\mathbb R^n$ defined by $d_C(a,b) := \sigma_{C}(a-b)$. Define a function $G:\mathbb R^n \times \mathbb R^n \to \mathbb R$ by $$ G(x,y) := \sup_{b \in B}\inf_{a \in A}a^\top x - b^\top y + d_C(a,b). $$

Question. (1) Is the function $G$ known in the literature ? (2) Can it be computed explicitly / analytically for certain choices of $A$, $B$, and $C$ ?

It's not even clair what $G(x,x) = \sup_{b \in B}\inf_{a \in A}\;x^\top(a-b) + d_C(a,b)$ should correspond to.

Observations

  • Taking $x=y=0$, we see that if $X=(A,d_C)$ and $Y := (B,d_C)$, then $$ G(0,0) = \sup_{b \in B}\inf_{a \in A}d_C(a,b) = \sup_{b \in B} d_C(b,A) \le d_{\mbox{Hausdorff}}(X,Y), $$ with equality if $A=B$.
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    $\begingroup$ What do you mean by sum over points from s convex body, i.e., $\sum_{c \in C} c^{T}x$? $\endgroup$
    – Paata Ivanishvili
    Commented Feb 26, 2021 at 14:19
  • $\begingroup$ Typo: should be $\sup_c$ not $\sum_c$; fixed. $\endgroup$
    – dohmatob
    Commented Feb 26, 2021 at 14:25

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