The now-classical Santal\'o inequality for an integrable, nonnegative function $f$ on $\mathbb R^n$ states
$M(f):=\left(\int_{\mathbb R^n}f\right)\inf_z\left(\int_{\mathbb R^n}(\tau_zf)^\circ\right) \leq M(e^{-|x|^2}),$
where $\tau_z f(x)= f(x-z)$ and $f^\circ$ is the polar of a function. We approximate polarity by defining the $L^p$ Laplace transform of a function $f$, $p\in (0,1)$ as,
$$\mathcal L_p(f)(x) = \left(\int_{\mathbb R^n}f(y)^\frac{1}{p}e^{x\cdot y}dy\right)^\frac{p}{p-1}.$$
Under mild regularity assumptions on $f$, such as log-concavity or continuity, $\lim_{p\to 0^{+}}\mathcal L_p(f)(x/p)=f^{\circ}(x)$.
In this work, which is joint with Cordero-Erausquin and Fradelizi, we define the $L^p$ volume product of a nonnegative function $f$ as
\[
M_p(f):=\Big(\int_{\mathbb R^n}f \Big)\, \inf_z \Big(\int_{\mathbb R^n}{\mathcal L}_p (\tau_z f)(x/p)\, dx \Big)^{1-p}.
\]
Our main theorem is that $M_p(f_t)$ is increasing along the Fokker-Planck heat-semi group. This extends a recent result by Nakamura-Tsuji, who obtained the same monotonicity when $f$ is even and integrable. An immediate corollary is
$M_p(f)\leq M_p(e^{-|x|^2})$. Sending $p\to 0^+$, we also obtain that $M(f_t)$ is increasing in $t$. Perhaps even more interesting is the analysis for the infimum: the infimum may be zero if $p$ is not sufficiently close to $0$. We characterize exactly when this occurs by studying the Laplace transform of log-concave functions, using and elaborating on some ideas by Klartag. If the infimum is not zero, then it is obtained at a unique point, which we call the $p$th Laplace-Santal\'o point of $f$.
