First, we choose a smooth function with positive Hessian on $\Delta_{\lambda-\beta}^{\vee}$ whose gradients stay in $\Delta_{\nu-\beta}$, and cover $\Delta_{\nu-2\beta}$.

Figure 1:   The domain for the first step.    The set of gradients.

As a second step, we need a continuous strictly convex function $\tilde{\nu}$ on $\beta\Delta^{\vee}$ that is an approximation of the function $\nu$ with the following properties.

• •

  $\tilde{\nu}$ is piecewise smooth.

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  $\operatorname{Hess}\tilde{\nu}>0$ on the smooth pieces.

• •

  The gradients along $\operatorname{cone}\tau$ belong to a neighborhood of the corresponding vertex in ${\Delta_{\nu}}$ that are pairwise disjoint, and do not meet the $\beta$ neighborhood of the barycenter of ${\Delta_{\nu}}$.

• •

  For a vertex $w\in T$, the $w$-directional derivatives equal $-\nu(w)$ in the star neighborhood of $\beta w$ in the barycentric subdivision of $\beta T$.

Figure 2: The set of gradients of $\tilde{\nu}$.

We obtain a convex function on $(\mathbb{R}^{d})^{*}$ if we consider the lower hull of the ($d+1$)-dimensional Minkowski sum of graphs of the two functions.

Finally, we want to obtain a function that is smooth along the strata. As explained in § 3.2, we convolute with a kernel that depends on the position $x\in\Delta^{\vee}_{\lambda}$ as follows. Consider the quadratic form

$$Q(x)=\sum_{v\in S}Q_{v}(x)\ \text{ where }\ Q_{v}(x)=\frac{1}{\rho_{v}(x)}v^{2}$$

This quadratic form is non-degenerate because the $v$’s span $(\mathbb{R}^{d})^{*}$. It has a dominant summand if $x$ is close to a facet. Now our kernel will be a normalized $e^{1-Q(x)}$. Its support – the ellipsoid given by $Q(x)\leq 1$ – depends on the position as sketched in the figure.

Figure 3: The support of the mollifier.

The obtained function will have a positive Hessian along the strata, so that the Legendre dual function will be smooth along its corresponding strata. ∎