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The answer is negative as can be seen by putting together these two facts:

1. There is a bounded convex lower semicontinuous functions defined on a closed and convex subset of $\mathbb R^2$ that is not continuous. One example is here. For a large stock of examples, see this.

2. Every convex, lower semicontinuous and positively homogeneous function on $\mathbb R^n$ is a support function. Theorem 13.2 in Rockefellar's Convex Analysis.

For then by (1), let $f$ be a non-continuous bounded convex lower semicontinuous function on a closed subset $D$ of the plane. Extend $f$ to all of the plane by setting it to $+\infty$ outside $D$. This is still convex and lower semicontinuous and it's not continuous on its effective domain.

Finally, define $g$ on $\mathbb R^3$ as follows: $$ g(x,y,z) = \begin{cases} zf(x/z,y/z) &\text{if }z>0\\ +\infty & \text{if }z\le 0\text{ and }(x,y)\ne 0\\ 0 &\text{if }(x,y,z)=0. \end{cases} $$ Let $f$ this is convex and lower semicontinuous everywhere and not continuous on the effective domain, and it's positive homogeneous. By (2), $g$ is a support function.

The answer is negative as can be seen by putting together these two facts:

1. There is a bounded convex lower semicontinuous functions defined on a closed and convex subset of $\mathbb R^2$ that is not continuous. One example is here. For a large stock of examples, see this.

2. Every convex, lower semicontinuous and positively homogeneous function on $\mathbb R^n$ is a support function. Theorem 13.2 in Rockefellar's Convex Analysis.

For then by (1), let $f$ be a non-continuous bounded convex lower semicontinuous function on a closed subset $D$ of the plane. Extend $f$ to all of the plane by setting it to $+\infty$ outside $D$. This is still convex and lower semicontinuous and it's not continuous on its effective domain.

Finally, define $g$ on $\mathbb R^3$ as follows: $$ g(x,y,z) = \begin{cases} zf(x/z,y/z) &\text{if }z>0\\ +\infty & \text{if }z\le 0\text{ and }(x,y)\ne 0\\ 0 &\text{if }(x,y,z)=0. \end{cases} $$ This is convex and lower semicontinuous everywhere and not continuous on the effective domain, and it's positive homogeneous. By (2), $g$ is a support function.

The answer is negative as can be seen by putting together these two facts:

1. There is a bounded convex lower semicontinuous functions defined on a closed and convex subset of $\mathbb R^2$ that is not continuous. One example is here.

2. Every convex, lower semicontinuous and positively homogeneous function on $\mathbb R^n$ is a support function. Theorem 13.2 in Rockefellar's Convex Analysis.

For then by (1), let $f$ be a non-continuous bounded convex lower semicontinuous function on a closed subset $D$ of the plane. Extend $f$ to all of the plane by setting it to $+\infty$ outside $D$. This is still convex and lower semicontinuous and it's not continuous on its effective domain.

Finally, define $g$ on $\mathbb R^3$ as follows: $$ g(x,y,z) = \begin{cases} zf(x/z,y/z) &\text{if }z>0\\ +\infty & \text{if }z\le 0\text{ and }(x,y)\ne 0\\ 0 &\text{if }(x,y,z)=0. \end{cases} $$ Let $f$ this is convex and lower semicontinuous everywhere and not continuous on the effective domain, and it's positive homogeneous. By (2), $g$ is a support function.

The answer is negative as can be seen by putting together these two facts:

1. There is a bounded convex lower semicontinuous functions defined on a closed and convex subset of $\mathbb R^2$ that is not continuous. One example is here. For a large stock of examples, see this.

2. Every convex, lower semicontinuous and positively homogeneous function on $\mathbb R^n$ is a support function. Theorem 13.2 in Rockefellar's Convex Analysis.

For then by (1), let $f$ be a non-continuous bounded convex lower semicontinuous function on a closed subset $D$ of the plane. Extend $f$ to all of the plane by setting it to $+\infty$ outside $D$. This is still convex and lower semicontinuous and it's not continuous on its effective domain.

Finally, define $g$ on $\mathbb R^3$ as follows: $$ g(x,y,z) = \begin{cases} zf(x/z,y/z) &\text{if }z>0\\ +\infty & \text{if }z\le 0\text{ and }(x,y)\ne 0\\ 0 &\text{if }(x,y,z)=0. \end{cases} $$ Let $f$ this is convex and lower semicontinuous everywhere and not continuous on the effective domain, and it's positive homogeneous. By (2), $g$ is a support function.

The answer is negative as can be seen by putting together these two facts:

1. There are bounded convex lower semicontinuous functions defined on a closed and convex subset of $\mathbb R^2$ that are not continuous. Example here.

2. Every convex, lower semicontinuous and positively homogeneous function on $\mathbb R^n$ is a support function. Theorem 13.2 in Rockefellar's Convex Analysis.

For by (1), let $f$ be a non-continuous bounded convex lower semicontinuous function on a closed subset $D$ of the plane $K = \{ (x,y,z) \in\mathbb R^3 : z = 1 \}$. Extend $f$ to all of $K$ by setting it to $+\infty$ outside $D$. This is still convex and lower semicontinuous and it's not continuous on its effective domain.

Finally, extend $f$ to all of $\mathbb R^3$ as follows. Let $f(x,y,z)=+\infty$ if $z\le 0$ and $(x,y,z)\ne 0$. Let $f(0,0,0)=0$. Let $f(x,y,z)=zf(x/z,y/z,1)$ if $z>0$. This is still convex and lower semicontinuous on $\mathbb R^3$ and not continuous on the effective domain, and it's positive homogeneous. By (2), our $f$ is a support function.

The answer is negative as can be seen by putting together these two facts:

1. There is a bounded convex lower semicontinuous functions defined on a closed and convex subset of $\mathbb R^2$ that is not continuous. One example is here.

2. Every convex, lower semicontinuous and positively homogeneous function on $\mathbb R^n$ is a support function. Theorem 13.2 in Rockefellar's Convex Analysis.

For then by (1), let $f$ be a non-continuous bounded convex lower semicontinuous function on a closed subset $D$ of the plane. Extend $f$ to all of the plane by setting it to $+\infty$ outside $D$. This is still convex and lower semicontinuous and it's not continuous on its effective domain.

Finally, define $g$ on $\mathbb R^3$ as follows: $$ g(x,y,z) = \begin{cases} zf(x/z,y/z) &\text{if }z>0\\ +\infty & \text{if }z\le 0\text{ and }(x,y)\ne 0\\ 0 &\text{if }(x,y,z)=0. \end{cases} $$ Let $f$ this is convex and lower semicontinuous everywhere and not continuous on the effective domain, and it's positive homogeneous. By (2), $g$ is a support function.