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The claim is false in general. The example at not-concave given by George Lowther disproves it. Indeed, in that example, $n=2$ and $A$ is the union of the unit disk and a one-point set at distance $R>1$ from the origin, so that $\mu(A_\delta)=f(\delta):=\pi[(1+\delta)^2+\delta^2]$ for $\delta\in[0,(R-1)/2]$ and $\lambda(\partial A)=f'(0)=2\pi$. So, the ratio of the RHS of your proposed inequality (1) to its LHS is $$\frac{(1+\delta)^2}{(1+\delta)^2+\delta^2}<1$$ for $\delta\in(0,(R-1)/2]$.

The claim is false in general. The example given by George Lowther disproves it. Indeed, in that example, $n=2$ and $A$ is the union of the unit disk and a one-point set at distance $R>1$ from the origin, so that $\mu(A_\delta)=f(\delta):=\pi[(1+\delta)^2+\delta^2]$ for $\delta\in[0,(R-1)/2]$ and $\lambda(\partial A)=f'(0)=2\pi$. So, the ratio of the RHS of your proposed inequality (1) to its LHS is $$\frac{(1+\delta)^2}{(1+\delta)^2+\delta^2}<1$$ for $\delta\in(0,(R-1)/2]$.

The claim is false in general. The example at not-concave given by George Lowther disproves it. Indeed, in that example, $n=2$ and $A$ is the union of the unit disk and a one-point set at distance $R>1$ from the origin, so that $\mu(A_\delta)=f(\delta):=\pi[(1+\delta)^2+\delta^2]$ for $\delta\in[0,(R-1)/2]$ and $\lambda(\partial A)=f'(0)=2\pi$. So, the ratio of the RHS of your proposed inequality (1) to its LHS is $$\frac{(1+\delta)^2}{(1+\delta)^2+\delta^2}<1$$ for $\delta\in[0,(R-1)/2]$.

The claim is false in general. The example at not-concave given by George Lowther disproves it. Indeed, in that example, $n=2$ and $A$ is the union of the unit disk and a one-point set at distance $R>1$ from the origin, so that $\mu(A_\delta)=f(\delta):=\pi[(1+\delta)^2+\delta^2]$ for $\delta\in[0,(R-1)/2]$ and $\lambda(\partial A)=f'(0)=2\pi$. So, the ratio of the RHS of your proposed inequality (1) to its LHS is $$\frac{(1+\delta)^2}{(1+\delta)^2+\delta^2}<1$$ for $\delta\in(0,(R-1)/2]$.

The claim is false in general. The example at not-concave given by George Lowther disproves it. Indeed, in that example, the ratio of the RHS of (1) to its LHS is $$\frac{(1+\delta)^2}{(1+\delta)^2+\delta^2}<1$$ for small enough $\delta>0$.

The claim is false in general. The example at not-concave given by George Lowther disproves it. Indeed, in that example, $n=2$ and $A$ is the union of the unit disk and a one-point set at distance $R>1$ from the origin, so that $\mu(A_\delta)=f(\delta):=\pi[(1+\delta)^2+\delta^2]$ for $\delta\in[0,(R-1)/2]$ and $\lambda(\partial A)=f'(0)=2\pi$. So, the ratio of the RHS of your proposed inequality (1) to its LHS is $$\frac{(1+\delta)^2}{(1+\delta)^2+\delta^2}<1$$ for $\delta\in[0,(R-1)/2]$.