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I want to prove for a convex set that contains zero a point $x$ is interior iff the value of Minkowski functional in that point is strictly less than $1$. is there anyone to help me.

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Just write the definition of the minkowski functional... – Emanuele Paolini Feb 17 '13 at 16:53

Unless you make additional assumptions on the convex set, this is not true in general. Consider the set $$A = \{ (x,0) \colon x \in [-1,1] \}$$ in $\mathbb R^2$. The corresponding Minkowski functional $p_A$ evaluates to $p_A(x_0) = 1/2$ at $x_0 = (1/2, 0)$, yet $x_0$ is not an interior point of $A$ (since $A$ has no interior).

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