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Given an indicator function $1_X$ defined on a compact subset $X$ of a (potentially non commutative) group $G$ as $1_X(x) = 1$ if $x \in X$ and $1_X(x) = 0$ otherwise, and the involution $\widetilde{1}_X$ defined as $\widetilde{1}_X(x) = 1_X(x^{-1})$, define the auto-correlation as the group convolution

$1_{X} \star \widetilde{1}_X (x)$ = $\int_G 1_X(y) \ \widetilde{1}_X(y^{-1}x) \ d\mu(y)$

where the Haar measure $\mu$ on $G$ is left translation invariant i.e. $\mu(xS) = \mu(S)$ . Clearly the group identity element $id_G$ is in the support of the auto-correlation. I have a couple of questions. First, is $1_{X} \star \widetilde{1}_X (id_G)$ necessarily non-differentiable? Second, more generally are all the local maxima of the auto-correlation non-differentiable?

(edit: one can work out examples of indicator functions of disjoint unions of intervals in $\mathbb{R}$ or for simple convex shapes in $\mathbb{R}^2$ to verify the property in these special cases)

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# Group convolution of indicator functions

Given an indicator function $1_X$ defined on a compact subset $X$ of a (potentially non commutative) group $G$ as $1_X(x) = 1$ if $x \in X$ and $1_X(x) = 0$ otherwise, and the involution $\widetilde{1}_X$ defined as $\widetilde{1}_X(x) = 1_X(x^{-1})$, define the auto-correlation as the group convolution

$1_{X} \star \widetilde{1}_X (x)$ = $\int_G 1_X(y) \ \widetilde{1}_X(y^{-1}x) \ d\mu(y)$

where the Haar measure $\mu$ on $G$ is left translation invariant i.e. $\mu(xS) = \mu(S)$ . Clearly the group identity element $id_G$ is in the support of the auto-correlation. I have a couple of questions. First, is $1_{X} \star \widetilde{1}_X (id_G)$ necessarily non-differentiable? Second, more generally are all the local maxima of the auto-correlation non-differentiable?