Let us consider a Brownian motion on $[0,1]$ denoted by $(W_t)_{t \in [0,1]}$.
Let us consider the function $f_W(p) = \sup_{0 \le t \le 1 } (pt - W_t)$
What is the distribution of $f_W(p)$ for a fixed $p$?
More generally, what is the measure we obtain on the set of convex functions when we pushforward the Wiener measure on $C^0([0,1])$ by using the Legendre-Fenchel transform?
By Girsanov's theorem: Under the probability measure $Q$ with Radon-Nikodym density \begin{align} \frac{dQ}{dP}&=\exp\left(p\,W_1-\frac{p^2}{2}\right) \end{align} the process $(\widetilde{W}_t)_{t\in[0,1]}=(p\,t-W_t)_{t\in[0,1]}$ is a Brownian motion. Likewise, \begin{align} \frac{dP}{dQ}&=\exp\left(p\,\widetilde{W}_1-\frac{p^2}{2}\right)\,. \end{align}
Recall from [1] Proposition 2.8.1 that $$ Q[\widetilde{W}_t\in da,\widetilde{M}_t\in db]=\frac{2(2b-a)}{\sqrt{2\pi t^3}}\exp\left\{-\frac{(2b-a)^2}{2t}\right\}\,da\,db\,\quad a\le b, b\ge 0\,, $$ where $\widetilde{M}_t=\max\limits_{0\le s\le t}\widetilde{W}_s\,,$ $t\in[0,1]$.
Therefore,
\begin{align} &P[\widetilde{M}_1\le x]\\&=E_Q\left[\frac{dP}{dQ};\widetilde{M}_1\le x\right]\\ &=\int_0^x\int_{-\infty}^b\frac{dP}{dQ}(a)\cdot Q[\widetilde{W}_1\in da,\widetilde{M}_1\in db]\,,\\ &=\int_0^x\int_{-\infty}^b\frac{2(2b-a)}{\sqrt{2\pi}}\exp\left\{p\,a-\frac{p^2}{2}-\frac{(2b-a)^2}{2}\right\}\,da\,db\\ \tag{1} &=\int_0^x e^{2bp}\left[2\,p\,\Phi(b+p)-2p + \sqrt{2/\pi}\,\exp\left\{-\frac{(b+p)^2}{2}\right\}\right]\,db\\[3mm] &\boxed{=\Phi(x-p)-e^{2xp}\Phi(-x-p)\,,} \end{align} where $\Phi(x)$ is the standard normal cdf. Rewriting the integrand in (1) gives $$ P[\widetilde{M}_1\in dx]=e^{2xp}\Big[2\,p\,\Phi(x+p)-2p+2\varphi(x+p)\Big]\, $$ where $\varphi(x)$ is the standard normal density.
[1] Karatzas, Shreve, Brownian Motion and Stochastic Calculus.