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I have the following question which maybe is too naive.

Let $K$ be a convex cone on $\mathbf{R}^n$. Can we approximate $K$ by a sequence of polyhedral convex cone $K_i$ such that for any compactly supported continuous function $f$, the following holds?: $$ \lim_{i\to\infty} \int_{\partial K_i} f(x) dx = \int_{\partial K} f(x) dx $$ Here the integrals are taken with respect to the restriction of Lebesgue measure on the boundary of $K$ and $K_i$.

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