Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve.
Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$.

Let $\hat\gamma(t):=(\gamma(t),f(\gamma(t)))\in \mathbb{R}^{n+1}$ be the lift of $\gamma$ to the graph of $f$. Similarly on defines $\hat\gamma_i$ as the lift of $\gamma$ to the graph of $f_i$.

Question. is it true that $$length(\hat\gamma_i)\to length(\hat\gamma)?$$

Remark. I think I can prove it for $n=1$.

Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve.
Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$.

Let $\hat\gamma(t):=(\gamma(t),f(\gamma(t)))\in \mathbb{R}^{n+1}$ be the lift of $\gamma$ to the graph of $f$. Similarly on defines $\hat\gamma_i$ as the lift of $\gamma$ to the graph of $f_i$.

Question. is it true that $$length(\hat\gamma_i)\to length(\hat\gamma)?$$

Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve.
Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$.

Let $\hat\gamma(t):=(\gamma(t),f(\gamma(t)))\in \mathbb{R}^{n+1}$ be the lift of $\gamma$ to the graph of $f$. Similarly on defines $\hat\gamma_i$ as the lift of $\gamma$ to the graph of $f_i$.

Question. is it true that $$length(\hat\gamma_i)\to length(\hat\gamma)?$$

Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve.
Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$.

Let $\hat\gamma(t):=(\gamma(t),f(\gamma(t)))\in \mathbb{R}^{n+1}$ be the lift of $\gamma$ to the graph of $f$. Similarly on defines $\hat\gamma_i$ as the lift of $\gamma$ to the graph of $f_i$.

Question. is it true that $$length(\hat\gamma_i)\to length(\hat\gamma)?$$

Remark. I think I can prove it for $n=1$.