# SUPPLEMENTARY MATERIALS
Coherent Quantum Interconnection between On-Demand Quantum Dot Single Photons
and a Resonant Atomic Quantum Memory

Guo-Dong Cui(崔国栋)  G.-D. Cui, L. Schweickert, and K. D. Jöns contributed
equally to this work.  Department of Physics & Astronomy, Stony Brook
University, Stony Brook, NY 11794-3800, USA  Lucas Schweickert  G.-D. Cui, L.
Schweickert, and K. D. Jöns contributed equally to this work.  Department of
Applied Physics, Royal Institute of Technology, Albanova University Centre,
Roslagstullsbacken 21, 106 91 Stockholm, Sweden  Klaus D. Jöns
<EMAIL_ADDRESS>Department of Applied Physics, Royal Institute of
Technology, Albanova University Centre, Roslagstullsbacken 21, 106 91
Stockholm, Sweden  Institute for Photonic Quantum Systems (PhoQS), Center for
Optoelectronics and Photonics Paderborn (CeOPP), and Department of Physics,
Paderborn University, 33098 Paderborn, Germany  G.-D. Cui, L. Schweickert, and
K. D. Jöns contributed equally to this work.  Mehdi Namazi  Department of
Physics & Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA
Thomas Lettner  Katharina D. Zeuner  Lara Scavuzzo Montaña  Department of
Applied Physics, Royal Institute of Technology, Albanova University Centre,
Roslagstullsbacken 21, 106 91 Stockholm, Sweden  Saimon Filipe Covre da Silva
Marcus Reindl  Huiying Huang(黄荟颖)  Institute of Semiconductor and Solid State
Physics, Johannes Kepler University Linz, 4040, Austria  Rinaldo Trotta
Institute of Semiconductor and Solid State Physics, Johannes Kepler University
Linz, 4040, Austria  Dipartimento di Fisica, Sapienza Università di Roma,
Piazzale A. Moro 1, I-00185 Roma, Italy  Armando Rastelli  Institute of
Semiconductor and Solid State Physics, Johannes Kepler University Linz, 4040,
Austria  Val Zwiller  Department of Applied Physics, Royal Institute of
Technology, Albanova University Centre, Roslagstullsbacken 21, 106 91
Stockholm, Sweden  Eden Figueroa<EMAIL_ADDRESS>Department of
Physics & Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA
Brookhaven National Laboratory, Upton, NY, 11973, USA

##  I Experimental setup

The experiment setup of our hybrid quantum platform is demonstrated in Fig.
SM1. It consists of three main parts. The first part is the quantum-light
source, which generates on-demand pairs of entangled photons, with one photon
of the pair specifically tuned to generate quantum fields at the rubidium D1
line. In the second part, we shine the temporally short single photons onto a
room temperature rubidium ensemble, thereby coherently controlling the
temporal wave function of the short single photons by the on-resonance
interaction with the rubidium atoms. Lastly, after atomic interaction, we
detect the photons using a superconducting nanowire single-photon detector
(SNSPD).

Figure SM1: Combined setup to interface quantum dot single photons and a
rubidium light-matter interface. Upper left: Pulsed laser system for single
quantum dot excitation. Upper center quantum dot single photon production.
Upper right: spectral characterization of the quantum dot photons. Lower
center: quantum memory setup including rubidium room temperature ensemble.
Bottom right: single photon detection after interaction.

##  II Quantum dot sample

The quantum dot sample was grown by molecular beam epitaxy at the JKU in Linz.
First a bottom distributed Bragg reflector made of 9 pairs of
$\lambda$/4-thick Al0.95Ga0.05As ($68.9\text{\,}\mathrm{nm}$) and Al0.2Ga0.8As
layers is deposited as a mirror for a $\lambda$-cavity. The quantum dot layer
is located at the center of the $\lambda$-cavity made of a $\lambda$/2-thick
($123\text{\,}\mathrm{nm}$) layer of Al0.4Ga0.6As sandwiched between two
$\lambda$/4-thick ($59.8\text{\,}\mathrm{nm}$) Al0.2Ga0.8As layers. The
quantum dot layer is fabricated by Al-droplet etching Heyn  _et al._ (2009);
Huo  _et al._ (2014) on the Al0.4Ga0.6As layer followed by deposition of
$2\text{\,}\mathrm{nm}$ GaAs. The top mirror for the cavity is made of two
pairs of the same material combination as the bottom distributed Bragg
reflector. To protect the structure a 4 nm-thick GaAs protective layer covers
the final sample. The QD design emission wavelength is centered around $\sim$
$790\text{\,}\mathrm{nm}$ and a gradient in the mode position (Q factor of
about 50) is generated by stopping the substrate rotation during the top
Al0.2Ga0.8As cavity-layer deposition. The cavity design together with solid
immersion lens enhances the collection efficiency by $\sim$ 30 times compared
to an unstructured sample.

##  III Piezo-actuator integration

Figure SM2: Color-coded photoluminescence spectra of the XX transition tuned
in and out of resonance of rubidium D1 transition. The horizontal line is a
narrow-band cw laser locked to the Rubidium D1 line $5S_{1/2}F=1$
$\rightarrow$ $5P_{1/2}F^{\prime}=1$ transition as a reference.

To achieve stable and precise energy tuning of our quantum dot emission into
resonance with the rubidium $D_{1}$ $F=1\rightarrow F^{\prime}=1$ transition
we integrate the GaAs quantum dot structure on a gold coated PMN-PT
piezoelectric actuator (TRS technologies, thickness of
$200\,$\mathrm{\SIUnitSymbolMicro m}$$, $\langle 001\rangle$ orientation). The
quantum dot sample is mechanically thinned using diamond-based abrasive films
and transferred on the piezo-electric substrate with a bendable soft tip in
pick and place method, taking advantage of electrostatic forces. We use
cryogenic epoxy (Stycast, two component resin) resulting in a rigid connection
between the sample and the piezo-actuator with good thermal contact. More
details on the quantum dot sample transfer on the piezo-electric substrate can
be found in Ref. Zeuner  _et al._ , 2018. To verify the reversible emission
energy tuning we performed several piezo-actuator voltage sweeps. Fig. 2 in
the main text shows the tuning of the quantum dot emission by the application
of a DC voltage to the piezoelectric actuator, the emission is compared to a
laser locked to the rubidium D1 line $5S_{1/2}\,F=1$ $\rightarrow$
$5P_{1/2}\,F^{\prime}=1$ transition. Our device can be repeatably tuned in and
out of resonance with a much higher precision than the resolution of the
spectrometer [see Fig. SM2].

##  IV Effective Photon Frequency and Temporal Profiles

To characterize the spectrum of quantum dot photons with respect to the
desired Rb resonance, a high-resolution photoluminescence spectroscopy (HRPL)
setup is employed with the help of a tunable fiber-coupled Fabry-Pérot
interferometer that could resolve 70 MHz and has a free spectral range (FSR)
of 20 GHz. The measured spectral lineshape of the bi-exciton (XX) photon is
presented in Fig. SM3.

Figure SM3: Spectrum of quantum dot photon. The horizontal axis is the
detuning of the Fabry-Perot interferometer from 87Rb $D_{1}$ transition
$F=1\leftrightarrow F^{\prime}=1$ line in GHz. Green dots: transmitted photon
histograms through Fabry Perot interferometer at different resonance
frequencies. Red curve: fitted spectrum using two added Voigt intensity
profiles. One of them is the convolution of a Lorentzian with FWHM of 1.18 GHz
(corresponding to the 134 ps measured lifetime) and a Gaussian with a variance
of 1.31 GHz (corresponding to spectral diffusion broadening). The other,
centered at -4 GHz away, is the convolution of a Lorentzian with FWHM of 1.19
GHz and a Gaussian with a variance of 4.62 GHz (corresponding to the photon
sideband). Blue curve: the effective Gaussian spectrum used to capture the
dominant dynamics in our simulation. The FWHM is fitted to be 5.69 GHz using
one Gaussian profile. Insert: the effective Gaussian intensity profile versus
time, that is obtained via the inverse Fourier transformation from the
effective Gaussian spectrum. The FWHM is calculated to be 77.56 ps. It is used
as the input single-photon temporal profile in the quantum photon-atom
interaction simulation.

In order to perform our simulation of the single-photon dynamics, we use an
effective frequency spectrum of the quantum dot single photon as seen by the
atomic ensemble. This can be represented by a Gaussian profile, as indicated
in the blue profile presented in Fig. SM3. The effective line-shape width is
determined by a Gaussian fit to the measured spectrum. This follows the form:

$\displaystyle S_{\text{eff}}(\nu)\propto
e^{-4\ln(2)\frac{\nu^{2}}{\Delta\nu^{2}}}$ (1)

The full width at half maximum (FWHM) is $\Delta\nu=5.69$ GHz. From the
inverse Fourier transformation of the effective frequency spectrum, we get the
effective temporal profile of the photon in the time domain:

$\displaystyle\mathcal{I}_{\text{eff}}(t)\propto|\mathcal{E}_{\text{eff}}(t)|^{2}\propto\left|FT^{-1}\left[\sqrt{S_{\text{eff}}(\nu)}\right](t)\right|^{2}\propto
e^{-4\ln(2)\frac{t^{2}}{\Delta{t}^{2}}}$ (2)

where the photon’s FWHM in time is calculated to be $\Delta{t}=77.56$ ps. This
is displayed in the inset of Fig. SM3 and it is the input of our master
equation simulation to calculate the atomic dynamics.

##  V Maxwell-Bloch Equations Simulation of Light-Matter Interaction

We assume that the field transverses the atoms in $z$ direction with linear
polarization along $y$ direction. It is convenient to define positive- and
negative-frequency parts of the field.

$\displaystyle E(z):=E_{+}(z)+E_{-}(z)$ (3)

where $E_{+}(z)\simeq
i\sqrt{\frac{\hbar\omega_{p}}{2V\epsilon_{0}}}\sum_{k}a_{k}e^{ikz}$ and
$E_{-}(z)=E_{+}^{\dagger}(z)$. Here ${\omega_{p}}$ is the central frequency of
the quantum dot photon profile (probe). Since there is no external field, the
choice of quantization axis for an atom could be random. We define the
quantization axis also in $y$ direction, same as the photon linear
polarization axis. Thus the photon will induce $\pi$ transitions of the atoms
in this convention. In the Schrödinger picture, the Hamiltonian of all the
atoms located at different position $z$ interacting with the same field can be
written as (the definition of the atomic sublevels is presented in Fig. 1 of
the main text).

$\displaystyle
H_{N}=\sum_{n=1}^{N}\hbar\left(\omega_{11}\sigma_{11}^{(n)}+\omega_{22}\sigma_{22}^{(n)}+\omega_{33}\sigma_{33}^{(n)}+\omega_{44}\sigma_{44}^{(n)}\right)-\sum_{n=1}^{N}d_{y}^{(n)}\left(E_{+}(z_{n})+E_{-}(z_{n})\right)+\sum_{k}\hbar\omega_{k}a_{k}^{\dagger}a_{k}$
(4)

where $E(z_{n})$ is the field value at the position of the $n$th atom.

To move everything into the interaction picture, we first consider the photon
field part. Choose
$H_{ph}^{\prime}:=\sum_{k}\hbar\omega_{k}a_{k}^{\dagger}a_{k}$. Field
operators in interaction picture is calculated via unitary operator
$U_{ph}(t)=e^{-iH_{ph}^{\prime}t/\hbar}$ and can be expressed as
$E_{+}(z,t)=U^{\dagger}_{ph}(t)E_{+}(z)U_{ph}(t):={\mathcal{E}}(z,t)e^{-i(\omega_{p}t-k_{p}z)},\quad
E_{-}(z,t)=E_{+}^{\dagger}(z,t)$ where $k_{p}:=\omega_{p}/c$ is the central
wave vector and a slowly varying envelope is assumed to be

$\displaystyle{\mathcal{E}}(z,t):\simeq
i\sqrt{\frac{\hbar\omega_{p}}{2V\epsilon_{0}}}\sum_{k}a_{k}e^{-i(\delta\omega_{k}\,t-\delta
k\,z)}$ (5)

where $\delta\omega_{k}:=\omega_{k}-\omega_{p}$, and $\delta k:=k-k_{p}$. For
the picture transformation for the $n$th atom we adopt a spatially rotating
unitary operator
$U_{atom}^{(n)}(z,t)=e^{-\frac{i}{\hbar}H_{atom}^{\prime(n)}(t-\frac{z_{n}}{c})}$,
with unperturbed atomic Hamiltonian
$H_{atom}^{\prime(n)}:=\hbar\omega_{p}\left(\sigma_{33}^{(n)}+\sigma_{44}^{(n)}\right)$.
Thus the picture transformation operator for the whole system could be defined
as

$\displaystyle
U_{N}(t)=\prod_{n=1}^{N}\otimes\left[\left(\sigma_{11}^{(n)}+\sigma_{22}^{(n)}\right)+e^{-i(\omega_{p}t-k_{p}z_{n})}\left(\sigma_{33}^{(n)}+\sigma_{44}^{(n)}\right)\right]\otimes
e^{-i\sum_{k}\omega_{k}a_{k}^{\dagger}a_{k}t}$ (6)

The dipole moment operator is
$d^{(n)}_{y}(t)=\left(d_{y}^{(31)}\sigma_{31}^{(n)}+d_{y}^{(41)}\sigma_{41}^{(n)}+d_{y}^{(32)}\sigma_{32}^{(n)}+d_{y}^{(42)}\sigma_{42}^{(n)}\right)e^{i(\omega_{p}t-k_{p}z_{n})}+h.c.$
in the interaction picture. Similarly, the Hamiltonian of the whole system is
defined by
$H_{N}(t)=U_{N}^{\dagger}(t)\left(H_{N}-\sum_{n}H_{atom}^{\prime(n)}-H^{\prime}_{ph}\right)U_{N}(t)$,
from which one gets

$\displaystyle H_{N}(t)=$
$\displaystyle\sum_{n=1}^{N}\hbar\left(\omega_{21}\sigma_{22}^{(n)}-\Delta_{p}\sigma_{33}^{(n)}+(\omega_{43}-\Delta_{p})\sigma_{44}^{(n)}\right)$
(7)
$\displaystyle-\sum_{n=1}^{N}\left(d_{y}^{(31)}\sigma_{31}^{(n)}+d_{y}^{(41)}\sigma_{41}^{(n)}+d_{y}^{(32)}\sigma_{32}^{(n)}+d_{y}^{(42)}\sigma_{42}^{(n)}\right)\,\mathcal{E}(z_{n},t)+H.c.$

Here $\Delta_{p}:=\omega_{p}-\omega_{31}$ is the detuning between the photon
carrier frequency and the $F=1\leftrightarrow F^{\prime}=1$ transition.
$\hat{\sigma}_{ij}:=\ket{i}\bra{j}$ represent the atomic operators. We use
dipole moment matrix elements Brink and Satchler (1968)
$d_{y}^{(31)}:=\frac{1}{2\sqrt{3}}d_{\text{fine}}$,
$d_{y}^{(41)}:=\frac{\sqrt{5}}{2\sqrt{3}}d_{\text{fine}}$,
$d_{y}^{(32)}:=\frac{\sqrt{5}}{2\sqrt{3}}d_{\text{fine}}$ and
$d_{y}^{(42)}:=\frac{\sqrt{5}}{2\sqrt{3}}d_{\text{fine}}$, with
$d_{\text{fine}}=3.588\times 10^{-29}$ C$\cdot$m being the fine transition
dipole moment of D1 line. The continuous limit of the Hamiltonian is

$\displaystyle H(z,t)=$
$\displaystyle\hbar\,\omega_{21}\sigma_{22}-\hbar\,\Delta_{p}\sigma_{33}+\hbar\,(\omega_{43}-\Delta_{p})\sigma_{44}$
(8)
$\displaystyle-\left(d_{y}^{(31)}\sigma_{31}+d_{y}^{(41)}\sigma_{41}+d_{y}^{(32)}\sigma_{32}+d_{y}^{(42)}\sigma_{42}\right)\,\mathcal{E}(z,t)+H.c.$

In the above expression, a position-independent term like
$\omega_{31}\sigma_{22}$ means that this term is a common property that any
atom in the ensemble has regardless of its position, while a position-
dependent term like $\mathcal{E}(z,t)$ gives specific value for an atom at
that position. The many-body Hamiltonian in Eq.(7) is thus mapped into its
continuous limit in Eq.(8). Similarly, the dipole moment operator in
continuous limit becomes
$d_{y}(z,t)=\left(d_{y}^{(31)}\sigma_{31}+d_{y}^{(41)}\sigma_{41}+d_{y}^{(32)}\sigma_{32}+d_{y}^{(42)}\sigma_{42}\right)e^{i(\omega_{p}t-k_{p}z)}+H.c.$.

The time evolution of atomic ensemble is governed by the Lindblad master
equation Fleischhauer  _et al._ (2005); Breuer and Petruccione (2007):

$\frac{\partial{\rho(z,t)}}{\partial
t}=-\frac{i}{\hbar}[H(z,t),\rho(z,t)]+\sum_{eg}\frac{\Gamma_{eg}}{2}\left(2\sigma_{ge}\rho(z,t)\,\sigma_{eg}-\sigma_{ee}\rho(z,t)-\rho(z,t)\,\sigma_{ee}\right)$
(9)

where $e=3,4$ are excited states, $g=1,2$ are ground states. We use the atomic
polarization decay rates
$(\Gamma_{31},\Gamma_{32},\Gamma_{41},\Gamma_{42})=(\frac{1}{6},\frac{5}{6},\frac{1}{2},\frac{1}{2})\,\Gamma$,
and the spontaneous emission rate $\Gamma=2\pi\times 5.746\text{ MHz}$.

Meanwhile, the propagation of the single photon field slow envelope is
governed by the Maxwell’s equation

$\displaystyle\left(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\right)E(z,t)$ $\displaystyle=$
$\displaystyle\mu_{0}\frac{\partial^{2}}{\partial t^{2}}P_{y}(z,t)$ (10)

The left-hand-side (LHS) of the equation can be expressed as
$i\frac{2\omega_{p}}{c^{2}}\left(\frac{\partial{\mathcal{E}}(z,t)}{\partial
t}+c\frac{\partial{\mathcal{E}}(z,t)}{\partial
z}\right)e^{-i(\omega_{p}t-k_{p}z)}+H.c.$ in the slow envelope approximation.
The right-hand-side (RHS) of the equation can also be simplified noticing that
expectation of dipole moment operator is
$\braket{P_{y}(z,t)}:=n\,\text{Tr}[{d_{y}}(z,t)\,\rho(z,t)]=n\left(d_{y}^{(31)}\rho_{13}(z,t)+d_{y}^{(32)}\rho_{23}(z,t)+d_{y}^{(41)}\rho_{14}(z,t)+d_{y}^{(42)}\rho_{24}(z,t)\right)e^{i(\omega_{p}t-k_{p}z)}+H.c.$.
After comparing the same oscillating terms on both the LHS and RHS of
Maxwell’s equation, one gets first-order propagation of the single photon
field

$\displaystyle\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
z}\right){\mathcal{E}}(z,t)=i\frac{\omega_{p}n\,}{2\epsilon_{0}}\left[d_{y}^{(13)}\rho_{31}(z,t)+d_{y}^{(23)}\rho_{32}(z,t)+d_{y}^{(14)}\rho_{41}(z,t)+d_{y}^{(24)}\rho_{42}(z,t)\right]$
(11)

We use a Matlab routine to solve the coupled set of Maxwell-Bloch equations
(Lindbladian master equation (9) and the first order Maxwell’s equation (11)),
using as input the fitted effective Gaussian distribution in time
$\mathcal{E}_{\text{eff}}(t)$ and obtained all the elements of the atomic
density matrix. This provides full knowledge of the dynamics involved in the
light-matter interface.

##  VI Comparison of Simulation with Experimental Data

So far, our simulation has only taken into account the frequency spread of the
quantum dot photon. However, we have not considered the time uncertainty in
the emission of the photons, which is provided by the Wigner-Weisskopf theory
Weisskopf and Wigner (1930) applied to the bi-exciton transition. To
characterize the time uncertainty in the emission of the photon, we excite the
quantum dot with a pulsed laser at an 80 MHz repetition rate. After cross-
polarization filtering and transmission spectrometer, only the bi-exciton
photons are collected and time-tagged in SNSPD. Here we emphasize that there
is no atomic interaction before the measurement.

Figure SM4:  The temporal profile of the quantum dot photon. The black line
shows the EMG fitting of the measured photon temporal histograms from an SNSPD
measurement. The EMG is the convolution of an exponential decay profile
(dashed purple line, with a fitted lifetime of 134 ps) and a Gaussian function
(red solid line, FWHM 77.56 ps). An example photon temporal profile applied to
the simulation is marked in the yellow area plot.

Fig. SM4 shows the measured temporal envelop of the quantum dot photons (solid
black line). We fit the measured histogram using an exponentially modulated
Gaussian function (EMG)

$emg(t;t_{0},\Delta{t},\tau):=\frac{1}{2\tau}\exp{\left(-\frac{t-t_{0}}{\tau}+\frac{\Delta{t}^{2}}{16\ln(2)\,\tau^{2}}\right)}\leavevmode\nobreak\
\text{erfc}\left(-2\sqrt{\ln(2)}\frac{t-t_{0}}{\Delta{t}}+\frac{\Delta
t}{4\sqrt{\ln(2)\tau}}\right)$ (12)

The EMG represents the convolution of the effective Gaussian temporal envelope
of the photon together with a time uncertainty that has an exponential decay
form, with a fitted lifetime $\tau=134$ ps. The EMG-fitted Gaussian temporal
profile has a FWHM of $\Delta t=75.35$ ps, which is in good agreement with the
77.56 ps derived from the spectrum discussion in Section IV. The colored
Gaussians in Fig. SM4 represent photons arriving at different times weighted
by the exponentially decaying probability.

Figure SM5: (a-e) Benchmarking the convoluted simulations (thick dashed blue
lines) against the experimental data (thick red lines) at five temperatures.
The simulated dynamics of the photon envelops reaching the ensemble at
different times are represented by the individual thin lines, one of which is
highlighted in the yellow area for each temperature. The individual
simulations are weighted by the 134 ps exponential decay function following
their arrival times, then integrated to form the convoluted simulation. All
the simulations are conducted using an identical set of parameters except for
ensemble temperatures. The incident photon field amplitude is the only free
global parameter and is optimized for the multi-temperature fit between the
simulations and the data. The raw data are normalized to the $55^{\circ}$C
peak. (f) The red line represents the measured photon spectrum. The effective
Gaussian spectrum used in our simulation is shown in the blue line. The
horizontal axis represents the detuning from
${}^{87}\text{Rb}\leavevmode\nobreak\ D_{1}(F=1\leftrightarrow F^{\prime}=1)$
line.

We use these fits to account for the time uncertainty in the measured
histograms obtained after transmitting the photons through the atomic ensemble
(thick red lines in Fig. SM5 (a-e)). This is done by convoluting the simulated
dynamics (colored thin lines) with the fitted 134 ps exponential decay profile
mentioned above. The results of this procedure are shown in the dashed blue
lines for different ensemble temperatures. These results match well with the
measured histograms, justifying the procedure. Furthermore, we mention that
all the parameters remain identical among the simulations (defined as global
parameters), except for different ensemble temperatures (defined as local
parameters). The amplitude of the original photon envelope is the only free
global parameter, which is varied to obtain a multi-temperature fit that best
describes the experimental data globally.

In Fig. SM6, we show the inverse to the procedure described in Fig. SM5. We
compare the simulated photon dynamics (blue lines) against the deconvoluted
experimental histograms (red areas). These histograms are obtained from the
directly measured histograms (dashed yellow lines), deconvoluted with the 134
ps exponential decay profile (dashed purple lines), in order to remove the
time uncertainty of the spontaneous emission. Thus they correspond to the
photon temporal envelopes reaching the ensemble at a well-defined time, which
is the main input to the simulation. This is the procedure use to obtain the
figures in the main manuscript. The global agreement between the numerical
simulation and the experimental data enables us to make inferences about the
various physical phenomena mentioned in the main text.

Figure SM6: (a-k) Benchmarking the simulations (blue lines) against the
deconvoluted experimental data (red areas). The deconvoluted data at each
temperature is obtained from the measured histogram (dashed yellow line)
deconvoluted with the exponential decay (dashed purple line), and is
normalized to the 55∘C peak. The simulations are done with the same set of
global parameters for different temperatures. We use an effective Gaussian
temporal profile with 77.56 ps FWHM as the input field. The field amplitude is
the only free global parameter that is varied to fit all the simulations with
the data simultaneously. (l) The spectrum of the simulated effective photon is
shown in the blue curve and the measured one in the red line.

##  VII Ensemble energy accumulation

Once the atomic density matrix elements are obtained in our simulation, we
calculate the additional ensemble energy gained by the atomic ensemble after
interacting with the photons. We choose the ground state $\ket{1}$ as a
reference and define the total ensemble energy relative to that. The energy
stored in the atomic ensemble at a specific time $t$ is then defined as

$\displaystyle\text{Energy}(t)=\sum_{n=1}^{N}\hbar\left(\omega_{21}\rho_{22}^{(n)}(t)+\omega_{31}\rho_{33}^{(n)}(t)+\omega_{41}\rho_{44}^{(n)}(t)\right)$
(13)

where $N$ is the total number of atoms inside the crossed volume of the photon
path with the atomic ensemble, and $\rho_{ii}^{(n)}(t)\,(i=2,3,4)$ are the
diagonal density matrix elements of the $nth$ atom at a specific time $t$. For
convenience, the energy could also be defined in units of the photon energy
(quanta) $\hbar\omega_{p}$. We use this definition in Fig. 5 of the main text
that corresponds to $75^{\circ}$C, as well as Fig. SM7 that corresponds to all
the eleven temperatures.

As the energy accumulation is a function of the atomic populations, we show in
Fig. SM8 the simulated spatio-temporal evolution of the atomic populations for
the four levels as a function of time and propagation depth at 75
${}^{\circ}{C}$.

Figure SM7: Calculated ensemble energy accumulation using the simulated atomic
density matrix elements during and after the quantum dot photon passes through
the ensemble. Here we follow the definition of the relative ensemble energy in
Eq.(13). The calculation covers the 11 temperatures for which experiments have
been done. The energy gain is defined in units of $\hbar\omega_{p}$. It can be
shown that part of the photon energy is retained in the atomic ensemble after
it propagates through the cell. And this storage time is more than 20 times
longer than the effective photon time width.  Figure SM8:  Simulated spatio-
temporal evolution of the diagonal density matrix elements that represent
population evolution during- and after the interaction with quantum dot photon
when the cell temperature is at 75 ${}^{\circ}{C}$. (a-b) Population drop in
the ground states $\ket{F=1(2)}$ of $D_{1}$ transition compared with their
thermal equilibrium values before the photon enters. (c-d) Population increase
in the excited states $\ket{F^{\prime}=1(2)}$.

## References

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E. Milman’s result (see section 2.3). Namely $f$ is $1$-Lipschitz and such
that

$\mathrm{Var}_{\mu}(f)\approx\|f\|_{\infty}^{2}\approx C_{P}(\mu).$

By the decomposition of variance

$\mathrm{Var}_{\mu}(f)=\mathbb{E}\mathrm{Var}_{\mu_{t}}(f)+\mathrm{Var}\left(\int_{\mathbb{R}^{n}}f\,d\mu_{t}\right)$

For the first term we proceed in the same way as before: by improved
Lichnerowicz and since $f$ is $1$-Lipschitz, we have

$\mathbb{E}\mathrm{Var}_{\mu_{t}}(f)\leq\frac{\mathbb{E}\|A_{t}\|^{1/2}}{\sqrt{t}}.$

For the second term, we proceed differently. The process $M_{t}=\int
f\,d\mu_{t}$ is a martingale, whose derivative is

$dM_{t}=\left(\int_{\mathbb{R}^{n}}f(x)(x-a_{t})\,d\mu_{t}\right)\cdot
dW_{t}.$

Since $\|f\|_{\infty}^{2}\lesssim C_{P}(\mu)$ we get from Cauchy-Schwarz

$\left|\int_{\mathbb{R}^{n}}f(x)(x-a_{t})\,d\mu_{t}\right|^{2}\lesssim
C_{P}(\mu)\|A_{t}\|.$

Hence

$\mathrm{Var}\left(\int f\,d\mu_{t}\right)\lesssim
C_{P}(\mu)\int_{0}^{t}\mathbb{E}\|A_{s}\|\,ds.$

If $\mathbb{E}\|A_{t}\|\lesssim 1$ up to time $t_{0}$ we finally get

$C_{P}(\mu)\lesssim t^{-1/2}+t\cdot C_{P}(\mu),$

for all $t\leq t_{0}$, which indeed implies $C_{P}(\mu)\lesssim t_{0}^{-1/2}$
(assuming $t_{0}\leq 1$, say). One thing that we can notice from this proof is
that if we replace the improved Lichnerowicz inequality by the usual one,
namely $C_{P}(\mu)\leq 1/t$ in the $t$-uniformly log-concave case, we also get
something non trivial, namely

$C_{P}(\mu)\lesssim t_{0}^{-1}.$ (37)

This is obviously a lot worse than what we get from improved Lichnerowicz, but
still non trivial. As a matter of fact, all the aforementioned works on the
KLS conjecture (prior to the latest one by Klartag in which the improved
Lichnerowicz inequality is established) rely on this estimate, one way or
another. The other argument to get Poincaré from the bound on the conditional
covariance (see section 5) may be more elementary and more natural in a way,
but it only works if one happens to know the improved Lichnerowicz inequality.
If you combine it with the usual Lichnerowicz inequality you get nothing.
Now we define the constants $K_{n}$ and $S_{n}$ by

$K_{n}=\sup\left\\{\|\sum_{i=1}^{n}(\mathbb{E}X_{i}X^{\otimes
2})^{2}\|\right\\},\quad S_{n}=\sup\\{\frac{1}{n}\mathrm{Var}|X|^{2}\\}$

where both sup are taken over all log-concave isotropic random vectors on
$\mathbb{R}^{n}$. The constant $S_{n}$ is called the thin-shell constant. The
thin-shell conjecture asserts that the sequence $(S_{n})$ is bounded. This is
a weak form of the KLS conjecture as we only require Poincaré for a very
specific function, namely the Euclidean norm squared. It was mentioned in the
first section in connection with the central limit problem for convex sets. A
variant of what we have done above shows that in the isotropic log-concave
case we have $\mathbb{E}\|A_{t}\|\lesssim 1$ up until times $(CK_{n}\log
n)^{-1}$. Hence from (37)

$C_{n}\lesssim K_{n}\log n.$

Moreover, by definition of $S_{n}$, given a log-concave and isotropic vector
$X$ on $\mathbb{R}^{n}$, a unit vector $u$, and an orthogonal projection $P$
of rank $k$, we have $\mathbb{E}(X\cdot u)|PX|^{2}\leq\sqrt{kS_{k}}$. Applying
this to suitable chosen projections $P$ one can estimate the eigenvalues of
$\mathbb{E}(X\cdot u)X^{\otimes 2}$ and then arrive at the bound

$K_{n}\lesssim\sum_{k=1}^{n}\frac{S_{k}}{k}\lesssim S_{n}\log n.$

We refer to [26] for the details. Altogether this gives

$C_{n}\lesssim S_{n}(\log n)^{2}.$

In other words thin-shell implies KLS up to polylog. This was the original
result of Eldan.

##  8 Further localization results

###  8.1 An obstruction to a full solution of KLS

As we have seen above, the KLS conjecture would be implied by the following
statement: in the isotropic log-concave case the expected operator norm of
$\mathrm{Cov}(X\mid X+\sqrt{s}G)$ remains of order $1$ for all $s$.
Unfortunately such an estimate cannot be true as we shall see now.

Let $X=(X_{1},\dotsc,X_{n})$ be a random vector whose coordinates are iid and
such that $1+X_{i}$ is an exponential variable of parameter $1$. This is
clearly an isotropic log-concave vector on $\mathbb{R}^{n}$.

######  Proposition 64.

We have $\mathbb{E}\|\mathrm{Cov}(X\mid X+\sqrt{s}G)\|=O(1)$ for all $s\geq
C\log n$. On the other hand, if $s\leq\log n$ then
$\mathbb{E}\|\mathrm{Cov}(X\mid X+\sqrt{s}G)\|\geq cs$.

Note that from the tensorization property of the Poincaré inequality, we have
$C_{P}(X)=C_{P}(X_{1})$, in particular the Poincaré constant of $X$ does not
depend on $n$ and $X$ is not a counterexample to the KLS conjecture. Recall
also that we always have the bound

$\mathbb{E}\|\mathrm{Cov}(X\mid X+\sqrt{s}G)\|\leq s$

(acually this is true almost surely, not only in expectation). This examples
shows that this bound can be essentially sharp on a time range $[0,s_{0}]$
with $s_{0}\to\infty$, namely $s_{0}=\log n$. In particular at time $s_{0}$ we
have

$\mathbb{E}\|\mathrm{Cov}(X\mid X+\sqrt{s_{0}}G)\|\geq c\log n$

so the expected norm of the conditional covariance is not bounded for all
times. In view of this example, the best one could hope for is

$\mathbb{E}\|\mathrm{Cov}(X\mid X+\sqrt{s}G)\|\lesssim 1,\quad\forall s\geq
C\log n$ (38)

and for every log-concave isotropic $X$. Notice that there is still a gap
between this and the bound that we obtained in our theorem (in which $\log n$
is replaced by $\log^{2}n$). If true the estimate (38) would imply the bound

$C_{n}\lesssim(\log n)^{1/4},$

for the KLS constant. This seems to be the limit one could reach within this
framework. Going below this mark would have to rely on different arguments.

The proof of the proposition only relies on some analysis in one dimension.
Indeed since the coordinates of $X$ and $G$ are all independent it is clear
that the conditional law of $X$ given $X+\sqrt{s}G$ is just the $n$-fold
product of the law of $X_{1}$ condition on $X_{1}+\sqrt{s}G_{1}$. So the
conditional covariance is diagonal with i.i.d entries, and we just have to
estimate the expected maximum of these.

As a preliminary step, we need to compute the variance of a truncated
Gaussian.

######  Lemma 65.

Let $g$ be as standard Gaussian variable, then

$\mathrm{Var}(g\mid g\geq x)\approx\frac{1}{1+x_{+}^{2}}.$

###### Proof.

Observe that the conditional law of $g$ given $g\geq x$ is log-concave. Let us
use the ”How to think about 1D log-concave measures” proposition from the
first section. It implies in particular that for log-concave random variable
$X$ having density $f$ we have

$\|f\|_{\infty}^{2}\cdot\mathrm{Var}(X)\approx 1,$

where $f$ is the density of $X$. Applying this to the conditional law of $g$
given $g\geq x$ we get (which is indeed log-concave), we get
$\mathrm{Var}(g\mid g\geq
x)\approx\left(\int_{x}^{\infty}\mathrm{e}^{-y^{2}/2}\,dy\right)^{-2}$ if
$x\leq 0$ and

$\mathrm{Var}(g\mid g\geq
x)\approx\left(\mathrm{e}^{x^{2}/2}\int_{x}^{\infty}\mathrm{e}^{-y^{2}/2}\,dy\right)^{-2}$

for every positive $x$. The result follows easily. ∎

###### Proof of Proposition 64.

The conditional law of $X_{1}$ given $X_{1}+\sqrt{s}G_{1}$ is just a truncated
Gaussian. After some elementary computation we get

$\mathrm{Var}(X_{1}\mid X_{1}+\sqrt{s}G_{1})=s\cdot
v(\sqrt{s}-\frac{1}{\sqrt{s}}Y_{1}-G_{1})$ (39)

where $Y_{1}=X_{1}+1$ and $v$ is the function given by

$v(x)=\mathrm{Var}(G_{1}\mid G_{1}\geq x).$

Note that $Y_{1}$ is an exponential variable independent of $G_{1}$, hence

$\mathbb{P}(Y_{1}\geq s,G_{1}\geq 0)=\frac{1}{2}\mathrm{e}^{-s}.$

Since the function $v$ is bounded away from $0$ on $\mathbb{R}_{-}$, if
$Y_{1}\geq s$ and $G_{1}\geq 0$ then

$\mathrm{Var}(X_{1}\mid X_{1}+\sqrt{s}G_{1})\geq cs.$

As a result

$\mathbb{P}(\mathrm{Var}(X_{1}\mid X_{1}+\sqrt{s}G_{1})\geq
cs)\geq\frac{1}{2}\mathrm{e}^{-s}.$

By independence we get

$\mathbb{P}(\|\mathrm{Var}(X\mid X+\sqrt{s}G\|)\geq cs)\geq
1-(1-\frac{1}{2}\mathrm{e}^{-s})^{n}$

If $s\leq\log n$, the right-hand side is at least $1-\mathrm{e}^{-1/2}$. This
implies $\mathbb{E}\|\mathrm{Var}(X\mid X+\sqrt{s}G\|)\|\geq c^{\prime}s$ by
Markov inequality.
For the other inequality, since $v(x)\lesssim x^{-2}$ for large $x$, equation
(39) and the union bound imply in particular that if $C$ is a sufficiently
large constant

$\mathbb{P}(\|\mathrm{Cov}(X_{1}\mid X_{1}+\sqrt{s}G_{1})\|\geq
C)\leq\mathbb{P}(Y_{1}\geq\frac{s}{4})+\mathbb{P}(G_{1}\geq\frac{\sqrt{s}}{4})\leq
2\mathrm{e}^{-cs}.$

Hence, by the union bound again,

$\mathbb{P}(\|\mathrm{Cov}(X\mid X+\sqrt{s}G)\|\geq C)\leq
2n\mathrm{e}^{-cs}.$

Since $\|\mathrm{Cov}(X\mid X+\sqrt{s}G)\|\leq s$ almost surely this implies

$\mathbb{E}\|\mathrm{Cov}(X\mid X+\sqrt{s}G)\|\leq
2ns\cdot\mathrm{e}^{-cs}+C.$

This becomes $O(1)$ as soon as $s$ exceeds a sufficiently large multiple of
$\log n$. ∎

###  8.2 Concentration of measure

Recall from section 2 the definition of the concentration function of $\mu$:

$\alpha_{\mu}(r)=\sup\left\\{1-\mu(S_{r})\colon\mu(S)=1/2\right\\}.$

We saw in section 2 that log-concave measures satisfy exponential
concentration and moreover than the Poincaré constant and the exponential
concentration constant squared are of the same order. In particular the
current best estimate for KLS amounts to the following

$\alpha_{\mu}(r)\leq C\cdot\exp\left(-c\cdot\frac{r}{\sqrt{\log n}}\right).$
(40)

One of the points of this lecture is to show that one can go a bit beyond this
estimate.

######  Theorem 66.

If $\mu$ is log-concave and isotropic then its concentration function
satisfies

$\alpha_{\mu}(r)\leq
C\exp\left(-c\cdot\min\left(r,\frac{r^{2}}{\log^{2}n}\right)\right),\quad\forall
r\geq 0.$ (41)

We should make some comments on this result. First of all, the rate provided
by Theorem 66 is not smaller than that of (40) on the whole halfline. In
particular combining with E. Milman’s theorem would only lead to a $(\log
n)^{2}$ bound for the Poincaré constant of an isotropic log-concave measure
(rather than $\log n$). That being said, the theorem yields in particular the
rate $\mathrm{e}^{-cr}$, which is predicted by the KLS conjecture, as soon as
$r$ is larger than $\log^{2}n$ or so. As far as we know, this information
cannot be inferred from bound $C_{n}\lesssim\log n$ alone. Let us also mention
that one can prove the following variant of (41), in which the concentration
depends on the KLS constant $C_{n}$:

$\alpha_{\mu}(r)\leq C\exp\left(c\cdot\min\left(r,\frac{r^{2}}{C_{n}\cdot\log
n}\right)\right),\quad\forall r\geq 0.$ (42)

This inequality is taken from Bizeul [8], as is most of the material of this
lecture.

This concentration is reminiscent of Guédon-Milman [34] from 2011, where they
proved that every isotropic log-concave measure $\mu$ satisfies

$\mu\left(||x|-\sqrt{n}|\geq r\right)\leq
C\cdot\exp\left(c\cdot\min\left(r,\frac{r^{2}}{n^{2/3}}\right)\right),\quad\forall
r\geq 0.$

This is weaker than (41) in two ways, first of all the constant is much worse
($n^{2/3}$ vs $\log^{2}n$) and the deviation inequality is only for the
Euclidean norm, and not for every $1$-Lipschitz function, as in (41). This
application of stochastic localization to concentration dates back to Lee and
Vempala [53]. Their main result in that paper is the bound $C_{n}=O(n^{1/2})$
for the KLS constant, but they also obtain the inequality

$\alpha_{\mu}(r)\leq
C\cdot\exp\left(c\cdot\min\left(r,\frac{r^{2}}{n^{1/2}}\right)\right).$ (43)

In contrast with (42), they do not loose a logarithm when they pass from the
bound on the KLS constant to the deviation inequality. Their argument is very
delicate and clever but it only works with a polynomial estimate for $C_{n}$
and it does not allow to remove the logarithm from (42) now that we have a
logarithmic estimate for $C_{n}$.

The proof of Theorem 66 relies on the fact that uniformly log-concave measures
satisfy Gaussian concentration. This was already mentioned in section 2.4.

######  Proposition 67.

Let $\mu$ be a $t$-uniformly log-concave measure. Then for every measurable
set $S$ and every $r\geq 0$ we have

$\mu(S)(1-\mu(S_{r}))\leq\exp(-c\cdot tr^{2}),\quad\forall r\geq 0,$

where $c$ is a universal constant. In particular the Gaussian concentration
constant of $\mu$ is $O(t^{-1/2})$.

###### Proof.

We give a short proof based on the Prékopa-Leindler inequality: if $f,g,h$ are
non negative functions on $\mathbb{R}^{n}$ satisfying the inequality

$\sqrt{f(x)g(y)}\leq h(\frac{x+y}{2})$

for every $x,y\in\mathbb{R}^{n}$ then

$\sqrt{\int_{\mathbb{R}^{n}}f(x)\,dx\int_{\mathbb{R}^{n}}g(x)\,dy}\leq\int_{\mathbb{R}^{n}}h(x)\,dx.$

If $\mu$ is $t$-uniformly log-concave then its potential $V$ satisfies

$V(\frac{x+y}{2})\leq\frac{V(x)+V(y)}{2}-\frac{t}{8}|x-y|^{2}.$

Given a set $S$ and $\theta>0$, one can then see that the hypothesis of
Prékopa-Leinder applies to the functions
$f(x)=\mathbf{1}_{S}(x)\mathrm{e}^{-V(x)}$, $g(y)=\mathrm{e}^{\theta
d(y,S)-V(y)}$ and $h=\mathrm{e}^{-V+2\theta^{2}/t}$. From the conclusion of
Prékopa, we get

$\mu(S)\cdot\int_{\mathbb{R}^{n}}\mathrm{e}^{\theta
d(x,S)}\,d\mu\leq\mathrm{e}^{2\theta^{2}/t}.$

The conclusion then follows from Chernov inequality. ∎

One can be a bit more precise. As we already mentioned, in the Gaussian case
we know the exact value of the concentration function $\alpha_{\gamma_{n}}$.
Indeed, an integrated version of the isoperimetric inequality of Sudakov-
Tsirelson / Borell asserts that $\gamma_{n}(S_{r})$ is maximized when $S$ is a
halfspace. In particular

$\alpha_{\gamma_{n}}(r)=1-\Phi(r),\quad\forall r\geq 0.$

where $\Phi$ is the distribution function of the standard Gaussian variable.
Moreover, a deep result of Caffarelli asserts that a measure $\mu$ that is
more log-concave than a given Gaussian measure is the image of that Gaussian
by a $1$-Lipshitz map. Besides, it is clear that a pushforward by a
contraction can only lower the concentration function. As a result if $\mu$ is
$t$-uniformly log-concave then its concentration function is upper bounded by
that of the Gaussian measure with covariance $\mathrm{Id}/t$, namely we have

$\alpha_{\mu}(r)\leq 1-\Phi(\sqrt{t}\cdot r),\quad\forall r\geq 0.$

Since $1-\Phi(r)\leq\frac{1}{2}\mathrm{e}^{-r^{2}/2}$ for $r\geq 0$, this
implies Gaussian concentration. However this only improves upon Proposition 67
at the level of the value of the universal constant $c$, which is irrelevant
for our purposes.

###### Proof of Theorem 66.

Fix a set $S$ of measure $1/2$ and write

$1-\mu(S_{r})=\mathbb{E}(1-\mu_{t}(S_{r}))\leq\mathbb{E}(1-\mu_{t}(S_{r}))\mathbbm{1}_{\\{\mu_{t}(S)\geq
1/4\\}}+\mathbb{P}(\mu_{t}(S)\leq 1/4),$

where $(\mu_{t})$ is the stochastic localization of $\mu$. Since $\mu_{t}$ is
$t$-uniformly log-concave, the first term is at most $4\mathrm{e}^{-ctr^{2}}$,
by Proposition 67. To handle the second term recall that the martingale
$M_{t}:=\mu_{t}(S)$ satisfies

$dM_{t}=\left(\int_{S}(x-a_{t})\,d\mu_{t}\right)\cdot dW_{t}.$

Applying Cauchy-Schwarz we obtain

$\left|\int_{S}(x-a_{t})\,d\mu_{t}\right|^{2}\leq\mu_{t}(S)\|A_{t}\|\leq\|A_{t}\|.$

Hence the inequality

$\langle M\rangle_{t}\leq\int_{0}^{t}\|A_{s}\|\,ds.$

In particular if $\|A_{s}\|\leq 2$ on $[0,t]$ then $\langle M\rangle_{t}\leq
2t$. Therefore

$\mathbb{P}(M_{t}\leq\frac{1}{4})\leq\mathbb{P}(M_{t}\leq\frac{1}{4}\,\&\,\langle
M\rangle_{t}\leq 2t)+\mathbb{P}(\exists s\leq t\colon\|A_{s}\|\geq 2).$

We have seen in the previous lecture that the second term is at most
$\exp(-(Ct)^{-1})$, provided $t\leq(C\log^{2}n)^{-1}$. On the other hand since
$M_{0}=\mu(S)=1/2$, Freedman’s inequality (also from the previous lecture)
insures that

$\mathbb{P}(M_{t}\leq\frac{1}{4}\,\&\,\langle M\rangle_{t}\leq
2t)\leq\exp\left(-\frac{1}{C_{1}t}\right).$

Putting everything together we get

$\mu(\overline{S_{r}})\leq 4\exp(-c\cdot tr^{2})+2\exp(-(C_{2}t)^{-1})$

for every $t\leq(C\cdot\log n)^{-2}$. Choosing $t=\min(r^{-1},(C\cdot\log
n)^{-2})$ yields the result. ∎

Here is an example of an application of the theorem.

######  Corollary 68 (Paouris theorem [60]).

Suppose $\mu$ is log-concave and isotropic then

$\mu(|x|\geq r)\leq\exp(-cr),\quad\forall r\geq C\sqrt{n},$

where as usual $c,C$ are universal constants.

######  Remark 69.

The inequality can also be expressed in terms of moments. It asserts that if
$X$ is log concave and isotropic on $\mathbb{R}^{n}$ then the moments of the
Euclidean norm of $X$ remain constant for quite a while, namely

$(\mathbb{E}|X|^{p})^{1/p}\approx(\mathbb{E}|X|^{2})^{1/2}$

for $p$ as large as $\sqrt{n}$.

###### Proof.

We apply the concentration estimate to the $1$-Lipschitz function $f(x)=|x|$.
We get in particular

$\mu(|x|\geq m+r)\leq\mathrm{e}^{-cr},$

provided that $r\geq C\cdot\log^{2}n$, where $m$ is a median for $|x|$. Since
$m\leq 2\int|x|\,d\mu\leq 2\sqrt{n}$, the latest display is easily seen to
imply the desired inequality. ∎

Our main point here is that from the inequality (40) one can only recover the
Paouris inequality up to some logarithmic factor. That being said, the
inequality (41) is an overkill for this application, and the argument would go
through using (43) instead, and as a matter of fact this application to the
Paouris inequality is taken from Lee and Vempala’s paper [53].

###  8.3 Logarithmic Sobolev inequality and a variant of the KLS conjecture

Recall from section 2.5 that the log-Sobolev constant of probability measure
on $\mathbb{R}^{n}$ denoted $C_{LS}(\mu)$ is the best constant in the
inequality

$D(\nu\mid\mu)\leq\frac{1}{2}C_{LS}(\mu)I(\nu\mid\mu)$

where $D$ and $I$ denote the realtive entropy and Fisher information,
respectively. Once again, the uniformly log-concave case is well understood.

######  Theorem 70 (Bakry-Émery criterion).

If $\mu$ is $t$-uniformly log-concave then $C_{LS}(\mu)\leq t^{-1}$. The
inequality is sharp, equality is attained for the Gaussian measure of
covariance $t^{-1}\cdot\mathrm{Id}$.

There are many ways to prove this inequality, see for instance [21] for an
overview. Again we are interested in the log-concave case. However, in
contrast with the Poincaré inequality, not every log-concave measure satisfies
log Sobolev. Indeed, recall that log-Sobolev implies Gaussian concentration,
with explicit control of the constants, and that this can be reversed in the
log-concave case: the log-Sobolev constant and the Gaussian concentration
constant are actually of the same order for log-concave measures. To insure
log-Sobolev, one has to impose another condition on top of log-concavity, such
as having bounded support. The following result is due to Lee-Vempala [53].

######  Theorem 71.

Suppose $\mu$ is log-concave, isotropic, and supported on a set of diameter
$D$. Then $C_{LS}(\mu)\lesssim D$.

Let us remark that because of the equivalence between log-Sobolev and Gaussian
concentration in the log-concave case, a log-concave measure supported on a
set of diameter $D$ trivially has $O(D^{2})$ log-Sobolev constant. Since the
diameter of the support of an isotropic measure is at least $\sqrt{n}$ the
theorem improves greatly upon the trivial bound in the isotropic case. It
should also be noted that for the uniform measure on the $\ell_{1}$ ball
rescaled to be isotropic, the diameter of the support and the log-Sobolev
constant both are of order $n$.

###### Proof.

This is actually an easy consequence of our concentration result Theorem 66.
Indeed, the latter asserts that if $\mu$ is log concave and isotropic then

$\alpha_{\mu}(r)\leq
C\cdot\exp\left(-c\cdot\min(r,\frac{r^{2}}{\log^{2}n})\right),\quad\forall
r\geq 0.$

On the other hand if $\mu$ is supported on a set of diameter $D$ then
trivially $\alpha_{\mu}(r)=0$ if $r>D$. On the interval $[0,D]$ we have $r\leq
r^{2}/D$, and since $D\geq\sqrt{n}\geq\log^{2}n$, we finally obtain

$\alpha_{\mu}(r)\leq C\cdot\exp\left(c^{\prime}\cdot\frac{r^{2}}{D}\right).$

The Gaussian concentration constant is thus $O(D)$, which implies the desired
inequality by E. Milman’s result. ∎

Let us try to relax the bounded support assumption. We know that log-Sobolev
implies Gaussian concentration. In particular linear functions should have
sub-Gaussian tails, at a rate controlled by the log-Sobolev constant. Let us
be a bit more precise.

######  Definition 72.

Suppose $f$ is a function having mean zero for $\mu$. We denote by
$\|f\|_{\psi_{2}(\mu)}$ the Orlicz norm of $f$ associated to the Orlicz
function $\mathrm{e}^{r^{2}}-1$, namely the best constant $C$ in the
inequality

$\mu(|f|\geq r)\leq 2\cdot\exp\left(-\frac{r^{2}}{C^{2}}\right).$

The discussion above shows that for any probability measure and any direction
$\theta$ we have

$\|x\cdot\theta\|_{\psi_{2}(\mu)}^{2}\lesssim C_{LS}(\mu).$

It is natural to conjecture that this inequality could be reversed in the log-
concave case. This amounts to saying that the log-Sobolev constant is the
largest $\psi_{2}$-norm squared of a linear function, very much like the KLS
conjecture predicts that the Poincaré constant of a log-concave measure is up
to a constant the largest $L^{2}$-norm squared of a linear function.

######  Definition 73 (Log-Sobolev version of KLS constant, Bizeul [8]).

Let $D_{n}$ be the largest log-Sobolev constant of a log-concave measure for
which $\|x\cdot\theta\|_{\psi_{2}(\mu)}\leq 1$ for every direction $\theta$.

######  Conjecture 74 (Log-Sobolev KLS conjecture, Bizeul [8]).

$D_{n}=O(1).$

It follows from some result of Bobkov [12] from 2007 that $D_{n}=O(n)$. Using
stochastic localization, one can show the following.

######  Theorem 75 (Bizeul (2023)).

$D_{n}=O(n^{1/2}).$

###### Proof.

The idea is to combine Theorem 66 with a rather crude net argument. By E.
Milman’s theorem it is enough to prove that if $\mu$ log-concave is $\psi_{2}$
with norm $1$ in all directions, then its concentration function satisfies

$\alpha_{\mu}(r)\leq C\mathrm{e}^{-cr^{2}/\sqrt{n}}.$ (44)

Note that the $\psi_{2}$ norm is larger than the $L^{2}$ norm, maybe up to a
constant. So the covariance of $\mu$ has operator norm $O(1)$. The
concentration function of $\mu$ thus satisfies

$\alpha_{\mu}(r)\leq C\exp\left(-c\min(r,\frac{r^{2}}{\log^{2}n})\right),$

for every $r>0$. Here there is a small gap which we can leave as an exercise:
show that having an upper bound for the concentration function of isotropic
log-concave $\mu$ of the form $\alpha_{\mu}\leq\alpha_{*}$ implies that if
$\mu$ is log-concave but not necessarily isotropic then
$\alpha_{\mu}(r)\leq\alpha_{*}(r/\sqrt{\|\mathrm{Cov}(\mu)\|})$. We thus get
an estimate that is smaller than our target concentration if
$r>cr^{2}/\sqrt{n}$, namely $r<C\sqrt{n}$. Therefore, it is enough to prove
(44) when $r$ is a sufficiently large multiple of $\sqrt{n}$. Moreover, by
Markov inequality we have

$\mu(|x|\geq 2\sqrt{n})\leq\frac{1}{4n}\int|x|^{2}\,d\mu\leq\frac{1}{4}.$

So if $S$ is a set of measure $1/2$ then $S$ intersects the ball of radius
$2\sqrt{n}$. If $r\geq 2\sqrt{n}$ this implies easily that
$S_{2r}\supset\\{|x|\leq r\\}$, hence

$\alpha_{\mu}(2r)\leq\mu(|x|>r).$

So it is enough to prove that $\mu(|x|>r)\leq\mathrm{e}^{-cr^{2}/\sqrt{n}}$
for $r\geq C\sqrt{n}$. Now recall the $\psi_{2}$ hypothesis: For every
direction $\theta$ and every $r$, we have

$\mu\\{|x\cdot\theta|>r\\}\leq 2\mathrm{e}^{-r^{2}}.$ (45)

It is well-known that there exists $1/2$-net of the unit sphere of cardinality
$5^{n}$ at most. Let $N$ be such a set. Since any element $x$ in the sphere is
at distance $1/2$ at most from a point of $N$ we have

$|x|\leq 2\max_{\theta\in N}\\{x\cdot\theta\\},$

for every $x\in\mathbb{R}^{n}$. Applying (45) to every $\theta$ in the net and
the union bound we get

$\mu(|x|>r)\leq 2\cdot 5^{n}\mathrm{e}^{-r^{2}/4}.$

If $r>C\sqrt{n}$ for a sufficiently large constant $C$, we deduce from this
inequality

$\mu(|x|>r)\leq\mathrm{e}^{-r^{2}/8}$

which is even better than what we needed. ∎

This proof seems to have lots of slack. In particular it certainly does not
look like the $\psi_{2}$ hypothesis was fully exploited. However, as far as
the log-Sobolev version of the KLS conjecture is concerned this is the best
result around, as of today.

##  9 Bourgain’s slicing problem

Consider a centrally-symmetric convex body $K\subseteq\mathbb{R}^{n}$ (i.e.
$K=-K$). The maximal function operator associated with $K$, defined for
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ via

$M_{K}f(x)=\sup_{r>0}\int_{K}f(x+ry)\frac{dy}{Vol_{n}(K)}.$

Bourgain [16] proved that $\|M_{K}\|_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})}\leq C$ for a universal constant $C>0$. This led him to
study on another question, seemingly innocent:

######  Question 76.

Let $n\geq 2$ and suppose that $K\subseteq\mathbb{R}^{n}$ is a convex body of
volume one. Does there exist a hyperplane $H\subseteq\mathbb{R}^{n}$ such that

$Vol_{n-1}(K\cap H)>c$ (46)

for a universal constant $c>0$?

This question is still not completely answered, and in the last four decades
it emerged as an “engine” for the development of the research direction
discussed in these lectures. It is shown in [45] that the bound (46) holds
true if we replace the universal constant $c$ on the right-hand side by
$c/\sqrt{\log n}$. This is the currently best known result in the general
case.

######  Theorem 77 (Hensley [35], Fradelizi [29]).

Let $K\subseteq\mathbb{R}^{n}$ be a convex body whose barycenter lies at the
origin. Let $X$ be a random vector distributed uniformly in $K$, and assume
that $Cov(X)$ is a scalar matrix. Then for any $\theta_{1},\theta_{2}\in
S^{n-1}$,

$Vol_{n-1}(K\cap\theta_{1}^{\perp})\leq C\cdot
Vol_{n-1}(K\cap\theta_{2}^{\perp})$

where $C>0$ is a universal constant. In fact, $C\leq\sqrt{6}$.

###### Proof.

Let $\theta\in S^{n-1}$ and denote

$\sigma=\sqrt{\mathbb{E}(X\cdot\theta)^{2}}=\sqrt{\mathrm{Cov}(X)\theta\cdot\theta},$

which is independent of $\theta$. Write

$\rho_{\theta}(t)=\frac{Vol_{n-1}(K\cap(t\theta+\theta^{\perp}))}{Vol_{n}(K)},$

the density of the random variable $X\cdot\theta$. By the Brunn-Minkowski
inequality, $\rho_{\theta}$ is a log-concave probability density. The log-
concave random variable $X\cdot\theta/\sigma$ has mean zero and variance one,
and its density is $x\mapsto\sigma\rho_{\theta}(x\sigma)$. According to
Proposition 4 above, for any $x\in\mathbb{R}$,

$c^{\prime}1_{\\{|x|\leq
c^{\prime\prime}\\}}\leq\sigma\rho_{\theta}(x\sigma)\leq Ce^{-c|x|}$

In particular, $c\leq\rho_{\theta}(0)\leq C$, for some universal constants
$c,C>0$. ∎

From this proof we may obtain a few more conclusions. First, that among all
hyperplane sections parallel to a given hyperplane, the hyperplane section
through the barycenter has the largest volume, up to a multiplicative
universal constant. Second, that when $K\subseteq\mathbb{R}^{n}$ is a centered
convex body of volume one, for any $\theta\in S^{n-1}$,

$Vol_{n-1}(K\cap\theta^{\perp})\cdot\sqrt{\mathbb{E}(X\cdot\theta)^{2}}\sim
1.$

Here $\theta^{\perp}=\\{x\in\mathbb{R}^{n}\,;\,x\cdot\theta=0\\}$ and we
abbreviate $A\sim B$ if $c\cdot A\leq B\leq C\cdot A$ for universal constants
$c,C>0$. This leads to the following conclusion:

######  Corollary 78.

Let $K\subseteq\mathbb{R}^{n}$ be a convex body of volume one and let $X$ be a
random vector distributed uniformly on $K$. Then,

$\sup_{H}Vol_{n-1}(K\cap H)\sim\frac{1}{\sqrt{\|\mathrm{Cov}(X)\|_{op}}},$

where the supremum runs over all hyperplanes $H\subseteq\mathbb{R}^{n}$.

We thus see that Bourgain’s slicing problem can be formulated as a question on
the relation between the covariance of a convex body and its volume. Note that
the logarithm of the volume of a convex body is the differential entropy of a
random vector $X$ that is distributed uniformly over the convex body. In
general, when the random vector $X$ has density $\rho$ in $\mathbb{R}^{n}$,
its differential entropy is

$Ent(X)=-\int_{\mathbb{R}^{n}}\rho\log\rho.$

######  Definition 79.

For a convex body $K\subseteq\mathbb{R}^{n}$ we define its isotropic constant
to be

$L_{K}=\left(\frac{\det\mathrm{Cov}(K)}{Vol_{n}(K)^{2}}\right)^{\frac{1}{2n}}$

where $Cov(K)$ is the covariance matrix of the uniform probability
distribution on $K$. More generally, the isotropic constant of an absolutely-
continuous, log-concave random vector $X$ in $\mathbb{R}^{n}$ is

$L_{X}=\left(\frac{\det\mathrm{Cov}(X)}{e^{2Ent(X)}}\right)^{\frac{1}{2n}}.$
(47)

The isotropic constant of a convex body $K\subseteq\mathbb{R}^{n}$ of volume
one governs the volumes of its hyperplane sections. From Corollary 78 we see
that when $Vol_{n}(K)=1$, there always exists a hyperplane section
$H\subseteq\mathbb{R}^{n}$ with

$Vol_{n-1}(K\cap H)\geq c/L_{K}.$

Moreover, if we additionally assume that $Cov(K)$ is a scalar matrix, then for
any hyperplane $H\subseteq\mathbb{R}^{n}$ through the barycenter of $K$,

$Vol_{n-1}(K\cap H)\sim\frac{1}{L_{K}}.$

The slicing problem thus asks whether $L_{K}$ is universally bounded from
above.

Remark on the definition of the isotropic constant in the log-concave case.
Some variants of this definition exist, sometimes one replaces $Ent(X)$ by
$-\log\sup\rho$ or by $-\log\rho(\mathbb{E}X)$ or by
$2\log\mathbb{E}\rho^{-1/2}(X)$, where $\rho$ is the density of $X$. These
variants differ at most by a multiplicative universal constant, because of the
following lemma:

######  Lemma 80.

Denoting $\psi=-\log\rho$, we have

$\psi(\mathbb{E}X)\leq Ent(X)\leq\inf\psi+n$

and

$\mathbb{E}e^{\frac{\psi(X)}{2}}\leq e^{\frac{\inf\psi}{2}+(\ln 2)n}.$

###### Proof.

We may assume that $\rho$ is continuous in $\mathbb{R}^{n}$ in order to
neglect boundary terms in the integration by parts below. Let
$y\in\mathbb{R}^{n}$. Then by Jensen’s inequality and by the fact that any
convex function lies above its tangent at $X$,

$\psi(\mathbb{E}X)\leq\mathbb{E}\psi(X)=Ent(X)=\mathbb{E}\psi(X)\leq\mathbb{E}\left[\psi(y)-\nabla\psi(X)\cdot(y-X)\right]=\psi(y)+n.$

Additionally,

$\mathbb{E}e^{\frac{\psi(X)}{2}}=e^{\frac{\psi(y)}{2}}\int_{\mathbb{R}^{n}}e^{-\frac{\psi(x)+\psi(y)}{2}}dx\leq
e^{\frac{\psi(y)}{2}}\int_{\mathbb{R}^{n}}e^{-\psi\left(\frac{x+y}{2}\right)}dx=2^{n}e^{\frac{\psi(y)}{2}}\int
e^{-\psi}=2^{n}e^{\frac{\psi(y)}{2}}.$

The lemma follows by taking the infimum over all $y\in\mathbb{R}^{n}$ in these
two inequalities ∎

It what follows we work with the definition (47). While here we are interested
only in the log-concave case, the definition makes sense for any absolutely-
continuous random vector $X$ with finite second moments in $\mathbb{R}^{n}$.
The isotropic constant measures the difference between two ways to measure the
“size” of a random vector: its entropy and its covariance. Here are some basic
properties of the isotropic constant:

  1. 1.

It is an affine invariant, $L_{T(X)}=L_{X}$ for any invertible linear-affine
map $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$.

  2. 2.

If $X_{1},X_{2}\in\mathbb{R}^{n}$ are independent log-concave random vectors,
then for
$X=(X_{1},X_{2})\in\mathbb{R}^{2n}\cong\mathbb{R}^{n}\times\mathbb{R}^{n}$,

$L_{X}=\sqrt{L_{X_{1}}L_{X_{2}}}.$

  3. 3.

For any dimension $n$ and an absolutely-continuous random vector $X$ with
finite second moments in $\mathbb{R}^{n}$,

$L_{X}\geq\frac{1}{\sqrt{2\pi e}},$

with equality when $X$ is Gaussian. Indeed, it is well-known that among all
random vectors with a fixed covariance in $\mathbb{R}^{n}$, the differential
entropy is maximal for the Gaussian distribution. The proof is short. Suppose
that $X$ is a random variable of mean zero, variance one and density $\rho$.
Then for $\gamma(x)=(2\pi)^{-1/2}\exp(-x^{2}/2)$ the standard Gaussian
density,

$\int_{\mathbb{R}}\rho\log\frac{\gamma}{\rho}\leq\int_{\mathbb{R}}f\left(\frac{\gamma}{\rho}-1\right)=0$

so

$Ent(X)=-\int\rho\log\rho\leq-\int\rho\log\gamma=-\int_{-\infty}^{\infty}\rho(x)\log\frac{e^{-|x|^{2}/2}}{\sqrt{2\pi}}dx=\log\sqrt{2\pi
e}.$

Exercise. Explain why it is not a coincidence that this universal constant
$\sqrt{2\pi e}$ is “the same number” from the asymptotics
$Vol_{n}(\sqrt{n}B^{n})^{1/n}\approx 1/\sqrt{2\pi e}$.

  4. 4.

Some examples:

$L_{[0,1]^{n}}=\frac{1}{\sqrt{12}},\qquad\qquad
L_{\Delta^{n}}=\frac{(n!)^{1/n}}{(n+1)^{(n+1)/(2n)}\sqrt{n+2}}\approx\frac{1}{e}.$

where $\Delta^{n}$ is a regular simplex in $\mathbb{R}^{n}$.

There are quite a few equivalent formulations and conditional statements,
relating the isotropic constant to classical conjectures and results:

  * •

If the isotropic constant is maximized for the cube among all centrally-
symmetric convex set, then the Minkowski lattice conjecture follows, see
Magazinov [56] and references therein. The Minkowski lattice conjecture
suggests that if $L\subseteq\mathbb{R}^{n}$ is a lattice of determinant one,
then each of its translates intersects the set

$\left\\{x\in\mathbb{R}^{n}\,;\,\prod_{i=1}^{n}|x_{i}|\leq\frac{1}{2^{n}}\right\\}.$

This was proven in two dimensions by Minkowski in 1908.

  * •

If the isotropic constant is maximized for the simplex among all convex
bodies, then the Mahler conjecture follows in the non-symmetric case. This
conjecture suggests that among all convex bodies $K\subseteq\mathbb{R}^{n}$,
the volume product

$Vol_{n}(K)\cdot Vol_{n}(K^{\circ})$

is minimized when $K$ is a centered simplex [43]. This was proven in two
dimensions by Mahler in 1908. Here

$K^{\circ}=\left\\{x\in\mathbb{R}^{n}\,;\,\forall y\in K,\ x\cdot y\leq
1\right\\}$

is the dual body. Recall that $(K^{\circ})^{\circ}=K$ when $K$ is a closed,
convex set containing the origin. The Bourgain-Milman inequality resolves this
conjecture up to a factor that is only exponential in the dimension. It states
that for any convex body $K\subseteq\mathbb{R}^{n}$ containing the origin,

$Vol_{n}(K)\cdot Vol_{n}(K^{\circ})\geq(c/n)^{n},$

for a universal constant $c>0$.

  * •

Suppose that $K\subseteq\mathbb{R}^{n}$ is a convex body. Is there an
ellipsoid $\mathcal{E}\subseteq\mathbb{R}^{n}$ with
$Vol_{n}(\mathcal{E})=Von_{n}(K)$ such that

$Vol_{n}(K\cap C\mathcal{E})\geq\frac{1}{2}\cdot Vol_{n}(K)$

where $C>0$ is a universal constant? This is an equivalent formulation of the
slicing problem.

Exercise. Prove the equivalence using reverse Hölder inequalities for
quadratic polynomials.

For any convex body $K\subseteq\mathbb{R}^{n}$, Milman’s ellipsoid theorem
provides an ellipsoid $\mathcal{E}\subseteq\mathbb{R}^{n}$ with

$Vol_{n}(K\cap C\mathcal{E})\geq c^{n}\cdot Vol_{n}(K).$

This suffices for developing the Milman ellipsoid theory, which contains the
quotient of subspace theorem and reverse Brunn-Minkowski and the Bourgain-
Milman inequality. See Pisier [62] and references therein. The slicing problem
is a conjectural strengthening of Milman’s ellipsoids.

We move on to discuss the $\sqrt{\log}$-bound for the isotropic constant, and
the relation to the Poincaré constant and the thin shell constants. We define

$\sigma_{n}=\sup_{X}\sqrt{Var(|X|^{2})/n}$

where the supremum ranges over all isotropic, log-concave random vectors $X$
in $\mathbb{R}^{n}$. By reverse Hölder inequalities for polynomials we may
show that $\mathrm{Var}(|X|^{2})/n\sim\mathrm{Var}(|X|)$, and hence
$\sigma_{n}$ is roughly the maximal width of the thin spherical shell that
captures most of the mass of an isotropic, log-concave random vector.

From Corollary 42 we know that,

$\sigma_{n}\leq\sup_{X}\sqrt{C_{P}(X)\cdot
4\mathbb{E}|X|^{2}/n}\leq\sup_{X}2\sqrt{C_{P}(X)}\leq C\sqrt{\log n}.$

Hence it remains to prove:

######  Theorem 81 (Eldan, K. ’10).

For any convex body $K\subseteq\mathbb{R}^{n}$,

$L_{K}\leq C\sigma_{n}.$

(In fact, it is shown in [27] that $L_{X}\leq C\sigma_{n}$ for any log-concave
random vector $X$ in $\mathbb{R}^{n}$, but for simplicity we confine ourself
here for the convex body case. The slicing problem for convex bodies and for
log-concave measures are known to be equivalent, as shown by Ball [4, 38].

While yesterday we studied Gaussian convolution, the proof of Theorem 81
utilizes the closely related Laplace transform. Let us fix an isotropic, log-
concave random vector $X$ with density $\rho$ in $\mathbb{R}^{n}$. Its
logarithmic Laplace transform is

$\Lambda(y)=\Lambda_{X}(y)=\log\mathbb{E}e^{X\cdot y}.$

Since a log-concave random vector has exponential moments, the logarithmic
Laplace transform is finite near the origin. In fact, it is smooth in the open
convex set $\Omega=\\{\Lambda<\infty\\}$. For $y\in\Omega$ we write $X_{y}$
for a random vector with density

$\rho_{y}(x)=\frac{\rho(x)e^{x\cdot y}}{e^{\Lambda(y)}}.$

It is again a log-concave random vector, not necessarily isotropic, and we
think of it as a tilted version of the random vector $X$. We comment that it
is possible to view tilts using projective transformations, this leads to
conditional statement that the strong slicing conjecture implies the Mahler
conjecture, see [43].

######  Lemma 82.

For any $y\in\Omega$,

$\nabla\Lambda(y)=\mathbb{E}X_{y},\qquad\nabla^{2}\Lambda(y)=\mathrm{Cov}(X_{y}),\qquad\nabla^{3}\Lambda(y)=\mathbb{E}(X_{y}-a_{y})^{\otimes
3},$

where $a_{y}=\mathbb{E}X_{y}$.

Lemma 82 is proven by direct computation; the logarithmic Laplace transform is
the cumulant generating function. We see from Lemma 82 that $\Lambda$ is
convex, even strongly-convex as its Hessian is positive definite. In
particular the gradient $\nabla\Lambda:\Omega\rightarrow\mathbb{R}^{n}$ is a
one-to-one map. Consider the “tilted determinant” function

$F(y)=\log\det\nabla^{2}\Lambda(y)=\log\det\mathrm{Cov}(X_{y}).$

It measures how the determinant of the covariance matrix changes when we tilt
the given distribution. Occasionally we may view $F$ as a function that is
defined only up to an additive constant. Write $[F]$ for the equivalence class
of $F$ under the equivalence relation “$F$ is equivalent to $G$ if and only if
$F-G$ is a constant function”.

######  Lemma 83.

The following bound holds pointwise in all of $\Omega$:

$(\nabla^{2}\Lambda)^{-1}\nabla F\cdot\nabla F\leq n\sigma_{n}^{2}.$ (48)

###### Proof.

Let us prove this bound first for $y=0$ using the isotropicity of $X$.
Recalling how to differentiate a determinant, we see that for any unit vector
$v\in S^{n-1}$,

$\partial_{v}F(0)=\mathrm{Tr}\left[(\nabla^{2}\Lambda)^{-1}(0)\cdot\partial_{v}\nabla^{2}\Lambda(0)\right]=\mathbb{E}(X\cdot
v)|X|^{2}\leq\sqrt{\mathbb{E}(X\cdot
v)^{2}\cdot\mathrm{Var}(|X|^{2})}\leq\sqrt{n}\sigma_{n}.$

By considering the supremum over all $v\in S^{n-1}$, we obtain the desired
bound at $y=0$.

In order to obtain the bound for any $y\in\Omega$ we may either make a
computation, or alternatively, think invariantly without computing anything,
as we now explain.

Define a Riemannian metric on $\Omega$ via the Hessian of the log-Laplace
transform $\Lambda$. We look at the Hessian metric $(\Omega,g)$, where the
scalar product of two tangent vectors $u,v\in
T_{x}\mathbb{R}^{n}\cong\mathbb{R}^{n}$ is

$g_{x}(u,v)=\nabla^{2}\Lambda(x)u\cdot v.$

The main observation is that the expression on the left-hand side of (48) is
the squared Riemannian length of the Riemannian gradient of the function
$F:\Omega\rightarrow\mathbb{R}$. We say that

$\mathcal{M}_{X}=(\Omega,g,[F])$

is the “Riemannian package” associated with $X$. This means that $(\Omega,g)$
is a Riemannian manifold and that $F$ is a function on $\Omega$ modulo an
additive constant. An isomorphism between two Riemannian packages is a
bijective map which is a Riemannian isometry and transforms correctly the
function modulo the additive constant.

What happens to the Riemannian package associated with $X$ when we do various
operations?

  * •

When we translate $X$, the Riemannian metric stays the same, as well as the
function $F$. We get the same Riemannian package.

  * •

Tilting $X$ and switching to $X_{y}$ yields an isomorphism of the two
Riemannian packages by translation by $y$: We translate $\Omega,g$ and $[F]$
by the vector $y\in\Omega$. Any translation corresponds to a tilt and vice
versa.

  * •

Applying an invertible linear transformation to $X$ induces an isomorphism of
the Riemannian packages. We apply a linear transformation and push forward
$\Omega,g$ and $[F]$. (See also the paragraph before the next lemma).

By the first and last items, we proved (48) at the point $y=0$ for any log-
concave random vector (not necessarily centered or isotropic). By the middle
item, we proved (48) also at all other points of $\Omega$. ∎

It makes sense to say that we think of $X$ as a random vector defined on an
abstract affine space, rather than on $\mathbb{R}^{n}$, and observe that the
Riemannian manifold $(\Omega,g)$ is well-defined, as well as the function
$F:\Omega\rightarrow\mathbb{R}$ modulo additive constants. What can we say
about balls in this Riemannian manifold?

######  Lemma 84.

Assume that $X$ is a centered, log-concave random vector in $\mathbb{R}^{n}$.
Then for any $r>0$,

$\frac{1}{2}\cdot\\{\Lambda\leq r\\}\subseteq B_{g}(0,\sqrt{r}).$

###### Proof.

Let $y\in\Omega$ satisfy $\Lambda(2y)\leq r$. We need to find a curve from $0$
to $y$ whose Riemannian length is at most $r$. Let us try a line segment:

$\displaystyle Length_{g}([0,y])$
$\displaystyle=\int_{0}^{1}\sqrt{\nabla^{2}\Lambda(ty)y\cdot
y}dt=\int_{0}^{1}\sqrt{\frac{d^{2}}{dt^{2}}\Lambda(ty)}dt$
$\displaystyle\leq\sqrt{\int_{0}^{2}(2-t)\frac{d^{2}}{dt^{2}}\Lambda(ty)dt\cdot\int_{0}^{1}\frac{1}{2-t}dt}$
$\displaystyle=\sqrt{\log
2}\cdot\sqrt{\Lambda(2y)-[\Lambda(0)+\nabla\Lambda(0)\cdot(2y)]}=\sqrt{\log
2}\cdot\sqrt{\Lambda(2y)}\leq\sqrt{r}.$

∎

Let $X$ be an isotropic random vector in $\mathbb{R}^{n}$, distributed
uniformly in a convex body $K\subseteq\mathbb{R}^{n}$. We need two estimates
for the proof of Theorem 81:

  1. (i)

First, we need to show that for $r=n/\sigma_{n}^{2}$,

$Vol_{n}(K)\geq e^{-n}\cdot Vol_{n}(B_{g}(0,\sqrt{r})),$

the Euclidean volume of the Riemannian ball. This is related to mass transport
in a simple case.

  2. (ii)

Second, we need to show that

$Vol_{n}(\\{\Lambda\leq r\\})^{1/n}\geq c\frac{r}{n}L_{K}.$

This is related to the Bourgain-Milman inequality.

###### Proof of Theorem 81.

Since $X$ is isotropic and log-concave, by (i), (ii) and Lemma 84,

$L_{K}=Vol_{n}(K)^{-1/n}\leq c\cdot Vol_{n}(B_{g}(0,\sqrt{r}))^{-1/n}\leq
c\cdot Vol_{n}(\\{\Lambda\leq r\\})^{-1/n}\leq
C\frac{n}{rL_{K}}=C\frac{\sigma_{n}^{2}}{L_{K}},$

and $L_{K}\leq C\sigma_{n}$. ∎

###### Proof of estimate (i):.

The function $F$ vanishes at the origin, and by Lemma 48 it is a Riemannian
Lipschitz function with Lipschitz constant at most $\sqrt{n}\sigma_{n}$.
Hence,

$|F|\leq n\qquad\textrm{in}\qquad B_{g}(0,\sqrt{r}).$

Consequently, for any $y\in B_{g}(0,\sqrt{r})$,

$e^{-n}\leq\det\nabla^{2}\Lambda(y)\leq e^{n}.$

We will use the fact that $\nabla\Lambda(y)=\mathbb{E}X_{y}\in K$ and that
$y\mapsto\nabla\Lambda(y)$ is one-to-one. Changing variables, we obtain

$Vol_{n}(K)\geq\int_{\nabla\Lambda(B_{g}(0,\sqrt{r}))}1dx\stackrel{{\scriptstyle``x=\nabla\Lambda(y)^{\prime\prime}}}{{=}}\int_{B_{g}(0,\sqrt{r})}\det\nabla^{2}\Lambda(y)\geq
e^{-n}\cdot Vol_{n}(B_{g}(0,\sqrt{r})).$

∎

###### Proof of estimate (ii):.

For any $y\in rK^{\circ}$,

$\Lambda(y)=\log\mathbb{E}e^{y\cdot X}\leq\log(e^{r})=r.$

Therefore,

$\\{\Lambda\leq r\\}\supseteq rK^{\circ}.$

By the Bourgain-Milman inequality,

$Vol_{n}(\\{\Lambda\leq r\\})^{1/n}\geq Vol_{n}(rK^{\circ})^{1/n}\geq
c\frac{r}{n}Vol_{n}(K)^{-1/n}=c\frac{r}{n}L_{K}.$

∎

We remark that the Bourgain-Milman inequality has several proofs, and in
particular it may be proven using more delicate analysis of the log-Laplace
transform as shown by Giannopoulos, Paouris and Vrisiou [31].

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