11institutetext:  School of Physics & Astronomy, Cardiff University, Cardiff
CF24 3AA, UK 22institutetext:  LLR (Laboratoire Leprince-Ringuet), CNRS, École
Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
33institutetext:  Centro de Estudios de Física del Cosmos de Aragón (CEFCA),
Plaza San Juan, 1, planta 2, E-44001, Teruel, Spain 44institutetext:  AIM,
CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris
Cité, 91191 Gif-sur-Yvette, France 55institutetext:  Univ. Grenoble Alpes,
CNRS, IPAG, 38000 Grenoble, France 66institutetext:  Institut d’Astrophysique
Spatiale (IAS), CNRS and Université Paris Sud, Orsay, France 77institutetext:
Institut Néel, CNRS and Université Grenoble Alpes, France 88institutetext:
Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, avenue des Martyrs,
38000 Grenoble, France 99institutetext:  Dipartimento di Fisica, Sapienza
Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy
1010institutetext:  Institut de RadioAstronomie Millimétrique (IRAM),
Grenoble, France 1111institutetext:  Centro de Astrobiología (CSIC-INTA),
Torrejón de Ardoz, 28850 Madrid, Spain 1212institutetext:  Instituto de
Radioastronomía Milimétrica (IRAM), Granada, Spain 1313institutetext:  Aix
Marseille Univ, CNRS, CNES, Laboratoire d’Astrophysique de Marseille,
Marseille, France 1414institutetext:  LERMA, Observatoire de Paris, PSL
Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, 75014
Paris, France 1515institutetext:  School of Earth and Space Exploration and
Department of Physics, Arizona State University, Tempe, AZ 85287
1616institutetext:  Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis
boulevard Arago, 75014 Paris, France 1717institutetext:  Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA 1818institutetext:  Univ. Toulouse, CNRS, IRAP, 9 Av.
du Colonel Roche, BP 44346, 31028, Toulouse, France

# GASTON: Galactic Star Formation with NIKA2

A new population of cold massive sources discovered

N. Peretto GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star
Formation with NIKA2<EMAIL_ADDRESS>A. Rigby GASTON: Galactic
Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 R. Adam
GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation
with NIKA2 P. Ade GASTON: Galactic Star Formation with NIKA2GASTON: Galactic
Star Formation with NIKA2 P. André GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 A. Andrianasolo GASTON:
Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2
H. Aussel GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star
Formation with NIKA2 A. Bacmann GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 A. Beelen GASTON: Galactic
Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 A. Benoît
GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with
NIKA2 A. Bideaud GASTON: Galactic Star Formation with NIKA2GASTON: Galactic
Star Formation with NIKA2 O. Bourrion GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 M. Calvo GASTON: Galactic Star
Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 A. Catalano
GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with
NIKA2 B. Comis GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star
Formation with NIKA2 M. De Petris GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 F.-X. Désert GASTON: Galactic
Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 S. Doyle
GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with
NIKA2 E. F. C. Driessen GASTON: Galactic Star Formation with NIKA2GASTON:
Galactic Star Formation with NIKA2 A. Gomez GASTON: Galactic Star Formation
with NIKA2GASTON: Galactic Star Formation with NIKA2 J. Goupy GASTON: Galactic
Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 F.
Kéruzoré GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star
Formation with NIKA2 C. Kramer GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 B. Ladjelate GASTON: Galactic
Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 G. Lagache
GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with
NIKA2 S. Leclercq GASTON: Galactic Star Formation with NIKA2GASTON: Galactic
Star Formation with NIKA2 J.-F. Lestrade GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 J.F. Macías-Pérez GASTON:
Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2
P. Mauskopf GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star
Formation with NIKA2GASTON: Galactic Star Formation with NIKA2GASTON: Galactic
Star Formation with NIKA2 F. Mayet GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 A. Monfardini GASTON: Galactic
Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 F. Motte
GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with
NIKA2 L. Perotto GASTON: Galactic Star Formation with NIKA2GASTON: Galactic
Star Formation with NIKA2 G. Pisano GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 N. Ponthieu GASTON: Galactic
Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 V. Revéret
GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with
NIKA2 I. Ristorcelli GASTON: Galactic Star Formation with NIKA2GASTON:
Galactic Star Formation with NIKA2 A. Ritacco GASTON: Galactic Star Formation
with NIKA2GASTON: Galactic Star Formation with NIKA2 C. Romero GASTON:
Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2
H. Roussel GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star
Formation with NIKA2 F. Ruppin GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 K. Schuster GASTON: Galactic
Star Formation with NIKA2GASTON: Galactic Star Formation with NIKA2 S. Shu
GASTON: Galactic Star Formation with NIKA2GASTON: Galactic Star Formation with
NIKA2 A. Sievers GASTON: Galactic Star Formation with NIKA2GASTON: Galactic
Star Formation with NIKA2 C. Tucker GASTON: Galactic Star Formation with
NIKA2GASTON: Galactic Star Formation with NIKA2 R. Zylka GASTON: Galactic Star
Formation with NIKA2GASTON: Galactic Star Formation with NIKA2

###### Abstract

Understanding where and when the mass of stars is determined is one of the
fundamental, mostly unsolved, questions in astronomy. Here, we present the
first results of GASTON, the Galactic Star Formation with NIKA2 large
programme on the IRAM 30m telescope, that aims to identify new populations of
low-brightness sources to tackle the question of stellar mass determination
across all masses. In this paper, we focus on the high-mass star formation
part of the project, for which we map a $\sim 2$ deg2 region of the Galactic
plane around $l=24^{\circ}$ in both 1.2 mm and 2.0 mm continuum. Half-way
through the project, we reach a sensitivity of 3.7 mJy/beam at 1.2mm. Even
though larger than our target sensitivity of 2 mJy, the current sensitivity
already allows the identification of a new population of cold, compact sources
that remained undetected in any (sub-)mm Galactic plane survey so far. In
fact, about 25% of the $\sim 1600$ compact sources identified in the 1.2mm
GASTON image are new detections. We present a preliminary analysis of the
physical properties of the GASTON sources as a function of their evolutionary
stage, arguing for a potential evolution of the mass distribution of these
sources with time.

##  1 Introduction

The characteristic shape of the initial mass function (IMF) of stars has its
origin rooted in the mechanisms leading to their formation. It is therefore
natural to envisage the existence of different star formation regimes for
every part of the IMF where the power-law slope changes. The nearly systematic
association of solar-mass prestellar cores with dense, self-gravitating
filaments arzoumanian2019 ; konyves2015  led to a picture where the peak of
the IMF would be the result of the gravitational fragmentation of such
filaments into individual cores, providing a compact and fixed mass reservoir
for the formation of low-mass stars inside them andre2010 ; andre2014 . For
high-mass stars, however, it is well known that thermal Jeans-type
fragmentation cannot explain the formation of cores more massive than a few
solar masses. Hence, the formation of massive stars requires additional
physics. There are strong indications that the dynamical evolution of the
parsec-scale mass reservoirs surrounding massive young stellar objects play a
key role. Even though the question about how massive stars form is still very
much debated, observations and simulations converge towards a picture where
massive stars form at the centre of globally collapsing clouds, quickly
growing in mass as a result of the large infall rates (e.g. bonnell2004, ;
wang2010, ; peretto2013, ; duartecabral2013, ). In this picture, massive stars
are said to be clump-fed, by opposition to low-mass star formation where stars
are core-fed. To understand how the IMF is determined across all stellar
masses, one now needs to focus on two additional mass intervals: Intermediate
mass stars ($2$ M${}_{\odot}<m_{*}<10$ M⊙) for which both core-fed and clump-
fed scenarios might be at work; and brown dwarfs ($m_{*}<0.1$ M⊙) for which
additional mechanisms such as dynamical instabilities of multiple stellar
systems bate2002  or disc fragmentation thies2010  might play a role in their
formation process. Unfortunately, no current (sub-)mm survey is either big nor
sensitive enough to detect significant populations of their corresponding
progenitors.

The NIKA2 NIKA2-Adam ; NIKA2-Calvo ; NIKA2-Bourrion  GASTON large programme
has been designed to fill that gap, with three key objectives. In addition to
tackling the question of the dominant formation mechanism of brown-dwarfs in
nearby clouds and mapping out a fraction of the Galactic plane at a mass
sensitivity and angular resolution never reached before in a multiple square
degree region, GASTON will, as a result of the dual frequency capability of
NIKA2, characterise dust properties across a wide range of environments. Here,
we present the first results of the Galactic plane part of the project,
roughly half-way through the observation campaign.

Figure 1: GASTON 1.2 mm map of the $\ell=24^{\circ}$ region of the Galactic
plane, cropped to show the approximate extent of the region that should reach
the target sensitivity. The grey filled circle in the bottom left corner of
the image matches the NIKA2 field-of-view size.

##  2 Observations

The section of the Galactic plane targeted within GASTON is
$\ell=[22.9^{\circ},24.9^{\circ}]$ and $b=[-0.6^{\circ},+0.7^{\circ}]$, which
has been selected for the high density of far-infrared sources identified by
Hi-GAL (molinari2016, ). The region is observed with a series of adjoining
0.5∘ by 1.5∘ rectangular fields that cross the mid-plane at a $\pm 30^{\circ}$
angle. Integrations are repeated over as many times as necessary to achieve an
approximately uniform sensitivity over the target area. A total area of $\sim
2.8$ deg2 is covered, the inner $\sim 1.8\leavevmode\nobreak\ $deg2 of which
will reach a uniform sensitivity. At the time of writing, about half of the
observations have been done (i.e. $\sim 35$h), and the measured sensitivities
are $\sigma_{1.2mm}\sim 3.7$ mJy/beam at 1.2 mm and $\sigma_{2.0mm}\sim 0.7$
mJy/beam at 2.0 mm. The angular resolutions are 12" and 18" at 1.2 mm and 2.0
mm, respectively. Flux calibration is regularly performed on primary
calibrators such as Uranus. Pointing errors are within 3". The data has been
reduced with the NIKA2 consortium IDL pipeline. The resulting 1.2 mm map is
shown in Fig. 1.

##  3 Compact source identification and cross-matching

As it can be seen in Fig. 1, structures with sizes of the order of the $\sim
6.5^{\prime}$ field-of-view are being recovered by the IDL reduction pipeline.
In order to identify compact sources, we filter out all structures larger than
an arcminute so that every source on the field is identified in a background-
free environment. To do this, we first convolve our GASTON 1.2mm image with a
Gaussian kernel to achieve an effective resolution of FWHM=60", and then
subtract that convolved image from the original GASTON image, effectively
filtering all sources larger than 60". We then run a dendrogram analysis
rosolowsky2008  on the filtered image to identify individual sources above
$3\sigma_{1.2mm}$ and compute their peak and integrated flux densities.

Figure 2: GASTON 1.2 mm compact source flux (left) and mass (right)
distributions. The blue histogram shows all GASTON sources identified. The red
histogram shows GASTON sources that are not associated with a band merged (BM)
Herschel clump elia2017 . The orange histogram shows GASTON sources that are
not associated with a Herschel 250 $\mu$m sources from molinari2016  Hi-GAL
catalogue. The vertical dashed line on the left panel shows the
3$\sigma_{1.2mm}$ detection threshold for beam-size sources.

In total, we have identified slightly over 1600 1.2 mm compact sources. The
integrated flux distribution of these sources is shown with the blue histogram
in Fig. 2-left. We can see that the histogram peaks at about 50 mJy, goes down
to $\sim 10$ mJy, and up to nearly 10 Jy for the brightest source. In order to
evaluate how many of these GASTON sources are already known, we cross-matched
their positions with catalogued Herschel sources, the most complete, and
sensitive, Galactic plane survey of cold dust structures available. For this,
we used two Herschel catalogues: the band merged catalogue (BM) from elia2017
, and the 250 $\mu$m catalogue of molinari2016 . By definition, the BM
catalogue is a subsample of the 250 $\mu$m catalogue. The histograms of GASTON
sources that do not have any counterpart in these two catalogues are also
displayed in Fig. 2, in which one can see that there are about 400 GASTON
sources ($\sim 25\%$) that do not appear in either catalogue. These sources
are, as expected, low-brightness sources (nearly all having a flux $\leq 0.2$
Jy) that were most likely not picked up in Herschel images as a result of the
lower far-infrared contrast of cold sources with cirrus noise. This sample of
previously undetected sources opens up a brand new window on the
characterisation of the earliest stages of star formation.

##  4 Source properties: Distance, dust temperature, and masses

One fundamental property of the GASTON sources is their mass. To estimate a
mass from dust continuum observations, one needs three quantities: flux,
distance, and dust temperature. Fluxes of GASTON sources have been already
presented in the previous section. In order to get a distance estimate for
each GASTON source we used the Galactic Ring Survey 13CO(1–0) data
romanduval2010  along with the Galactic rotation model from reid2016 . For
each source we get the corresponding 13CO(1–0) spectrum averaged over the
footprint of the GASTON source and, in case of multiple 13CO(1–0) velocity
components, we select the strongest one as the GASTON source velocity. We then
use the Bayesian distance estimator from reid2016  to determine the distance
of maximum likelihood. Finally, via a friend-of-friend algorithm, we group
nearby sources with similar velocities together, and use the median distance
of these sources as the single distance for that group. The idea behind this
is to mitigate the effects of peculiar velocities of clumps within clusters
being interpreted as differences in the distance.

Regarding the dust temperature of each GASTON source we used the same method
as developed by peretto2016 , and successfully used in combination with NIKA
data in rigby2018 . This method uses the ratio of the Herschel 160 $\mu$m flux
to the 250 $\mu$m flux as a proxy for temperature. Here, we further filter
both images using the same kernel as on the GASTON images to remove any large-
scale contaminating background emission. From the Herschel filtered images one
can get a dust temperature at 18" resolution (the resolution of the 250 $\mu$m
image) for every GASTON source. With distances, and dust temperatures one can
now compute the mass distributions of the GASTON compact sources. For this we
assumed the following specific dust opacity law
$\kappa_{\nu}=0.1\,(\nu/1\,\mathrm{THz})^{1.8}\,\mathrm{cm}^{2}\,\mathrm{g}^{-1}$,
which accounts for a 1% dust-to-gas mass ratio andre2010 ; rigby2018 . The
mass distributions of the three samples (see previous section) are shown in
Fig. 2-right. The striking feature of this figure is the fact that sources
that were not identified before in any other catalogue (orange histogram) have
a very similar mass distribution to rest of the sample, despite having lower
fluxes. This shows that this new sample is mostly made of cold compact and
relatively massive sources that were just not picked up by any other survey.
This source population could represent an important missing building block in
our understanding of the stellar mass determination.

##  5 Evolution of mass distributions

Figure 3: GASTON 1.2 mm compact source mass distributions as a function of
their infrared darkness parameter. The histograms show the mass distributions
of sources with: i) $f_{\mathrm{IRD}}=1$ (i.e. completely infrared-dark) in
orange; ii) $1>f_{\mathrm{IRD}}>0.5$ in pink; iii) $0.5>f_{\mathrm{IRD}}>0$ in
blue and iv) $f_{\mathrm{IRD}}=0$ (i.e. completely infrared-bright) in blue.
The differential distributions are displayed on the left-hand-side panel, the
cumulative forms are displayed in the right-hand-side panel.

As dusty, star-forming clumps evolve, embedded young-stars heat up an
increasing fraction of the clump dust mass. During that process, a cold clump
that was originally dark in the mid-infrared can become infrared bright. One
can therefore use the infrared darkness of a clump as a proxy for its
evolutionary stage battersby2017 . Using a combination of Spitzer 8 $\mu$m and
Herschel data, Watkins, Peretto et al. (in prep) have developed a method to
compute the fraction of a clump mass that is infrared dark. We here use that
method to characterise the infrared darkness of GASTON sources. We then split
our sample into four bins of infrared darkness parameters, from $f_{IRD}=1$
(totally infrared dark) to $f_{IRD}=0$ (totally infrared bright), and we plot
the mass distributions for each of the four subsamples. This is shown in Fig.
3. This figure tentatively shows that there are differences in the shape of
the mass distribution, with the most massive sources having intermediate
values of infrared darkness parameter. If one considers infrared darkness
$f_{IRD}$ as a valid proxy for time evolution, then this would suggest that
sources gain mass as they evolve from $f_{IRD}=1$ to $f_{IRD}=0.5$, and then
lose mass as they evolve towards $f_{IRD}=0$. This is what one expects in a
scenario where core and protostellar mass growth occur simultaneously
hatchell2008 .

##  6 Summary

We used NIKA2 on the IRAM 30m telescope to map over 2 square degrees of the
Galactic plane at 1.2 mm and 2.0 mm. While the observing campaign is still
ongoing we performed a first analysis of the 1.2 mm image. The high
sensitivity of these GASTON images allowed us to identify a brand new
population of low-brightness compact sources, which, being amongst the coldest
sources, have a similar mass distribution as those which were already known.
This new sample might well represent a missing building-block in our
understanding of the stellar mass determination process. Using infrared
darkness as a possible proxy of their evolutionary stage, we also suggest that
there might be some evidence for a time evolution of the clump mass
distribution. However, further detailed analysis of the data is required to
evaluate the robustness of such interpretation. The remaining integration time
of the GASTON field will allow us to increase source statistics on cold
intermediate-mass sources ($\sim 10$ M⊙), allowing the connection of our
results with those obtained in well-known nearby star-forming clouds. In this
context, the GASTON 2.0 mm data will be instrumental in confirming the nature
of low signal-to-noise ratio sources detected at 1.2 mm.

## Acknowledgements

N. P. and A. R. would like to thank STFC for support under the consolidated
grant number ST/N000706/1. We thank the Royal Society for providing computing
resources under Research Grant number RG150741. We would also like to thank
the IRAM staff for their support during the campaigns. The NIKA dilution
cryostat has been designed and built at the Institut Néel. In particular, we
acknowledge the crucial contribution of the Cryogenics Group, and in
particular Gregory Garde, Henri Rodenas, Jean Paul Leggeri, Philippe Camus.
This work has been partially funded by the Foundation Nanoscience Grenoble and
the LabEx FOCUS ANR-11-LABX-0013. This work is supported by the French
National Research Agency under the contracts "MKIDS", "NIKA" and
ANR-15-CE31-0017 and in the framework of the "Investissements dâavenirâ
program (ANR-15-IDEX-02). This work has benefited from the support of the
European Research Council Advanced Grant ORISTARS under the European Union’s
Seventh Framework Programme (Grant Agreement no. 291294). We acknowledge
fundings from the ENIGMASS French LabEx (R. A. and F. R.), the CNES post-
doctoral fellowship program (R. A.), the CNES doctoral fellowship program (A.
R.) and the FOCUS French LabEx doctoral fellowship program (A. R.). R.A.
acknowledges support from Spanish Ministerio de Economía and Competitividad
(MINECO) through grant number AYA2015-66211-C2-2.

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# Affine fractional Sobolev and isoperimetric inequalities

Julián Haddad  Departamento de Matemática, ICEx, Universidade Federal de Minas
Gerais, 30.123-970, Belo Horizonte, Brazil<EMAIL_ADDRESS>and  Monika
Ludwig  Institut für Diskrete Mathematik und Geometrie, Technische Universität
Wien, Wiedner Hauptstraße 8-10/1046, 1040 Wien, Austria
<EMAIL_ADDRESS>
###### Abstract.

Sharp affine fractional Sobolev inequalities for functions on $\mathbb{R}^{n}$
are established. For each $0<s<1$, the new inequalities are significantly
stronger than (and directly imply) the sharp fractional Sobolev inequalities
of Almgren and Lieb. In the limit as $s\to 1^{-}$, the new inequalities imply
the sharp affine Sobolev inequality of Gaoyong Zhang. As a consequence,
fractional Petty projection inequalities are obtained that are stronger than
the fractional Euclidean isoperimetric inequalities and a natural conjecture
for radial mean bodies is proved.

2020 AMS subject classification: 46E35 (35R11, 52A40)

##  1\. Introduction

Almgren and Lieb [1] established Pólya–Szegő inequalities for fractional
Sobolev norms. The equality case was settled by Frank and Seiringer [11]. A
consequence is the fractional Sobolev inequality: for $0<s<1$, there is
$\alpha_{n,s}>0$ such that for $f\in W^{s,1}(\mathbb{R}^{n})$,

(1)
$\|f\|_{{\frac{n}{n-s}}}\leq\alpha_{n,s}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{|x-y|^{n+s}}\,\mathrm{d}x\,\mathrm{d}y,$

and there is equality if and only if $f$ is a constant multiple of the
indicator function of a ball. Here $\|f\|_{p}$ denotes the $L^{p}$ norm of $f$
and $|\cdot|$ the Euclidean norm in $\mathbb{R}^{n}$ while
$W^{s,1}(\mathbb{R}^{n})$ denotes the fractional Sobolev space of $L^{1}$
functions with finite right side in (1). By a result of Bourgain, Brezis, and
Mironescu [3], for $f\in W^{1,1}(\mathbb{R}^{n})$,

(2) $\lim_{s\to
1^{-}}\alpha_{n,s}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{|x-y|^{n+s}}\,\mathrm{d}x\,\mathrm{d}y=\frac{1}{n\omega_{n}^{1/n}}\int_{\mathbb{R}^{n}}|\nabla
f(x)|\,\mathrm{d}x,$

where $\omega_{n}$ is the $n$-dimensional volume of the unit ball in
$\mathbb{R}^{n}$ and $W^{1,1}(\mathbb{R}^{n})$ denotes the Sobolev space of
$L^{1}$ functions $f$ with weak $L^{1}$ gradient $\nabla f$. Dávila [7] proved
a result corresponding to (2) for $f\in BV(\mathbb{R}^{n})$, the space of
$L^{1}$ functions with bounded variation. Hence, the sharp $L^{1}$ Sobolev
inequalities on $W^{1,1}(\mathbb{R}^{n})$ and $BV(\mathbb{R}^{n})$ follow from
the fractional Sobolev inequalities (1).

Gaoyong Zhang [38] established a sharp affine Sobolev inequality that is
significantly stronger than the classical $L^{1}$ Sobolev inequality. For
$n\geq 2$ and $C^{1}$ functions $f:\mathbb{R}^{n}\to\mathbb{R}$ with compact
support, Zhang [38] proved that

(3)
$\|f\|_{{\frac{n}{n-1}}}\leq\frac{\omega_{n}}{2\omega_{n-1}}\Big{(}\frac{1}{n}\int_{{\mathbb{S}}^{n-1}}\|\langle\nabla
f(\cdot),\xi\rangle\|_{1}^{-n}\,\mathrm{d}\xi\Big{)}^{-1/n}\leq\frac{1}{n\omega_{n}^{1/n}}\int_{\mathbb{R}^{n}}|\nabla
f(x)|\,\mathrm{d}x.$

Here $\langle\cdot,\cdot\rangle$ denotes the inner product and integration on
the $(n-1)$-dimensional unit sphere ${\mathbb{S}}^{n-1}$ is with respect to
the $(n-1)$-dimensional Hausdorff measure. The first inequality in (3) is now
called the affine Sobolev inequality or the Sobolev–Zhang inequality. It is
affine since both terms remain invariant under volume-preserving affine
transformations. Tuo Wang [35] extended the Sobolev–Zhang inequality to
$L^{1}$ functions of bounded variation and showed that there is equality in
this generalized Sobolev–Zhang inequality precisely for constant multiples of
indicator functions of ellipsoids.

The main aim of this paper is to establish affine fractional Sobolev
inequalities for $0<s<1$ that are stronger than the fractional Sobolev
inequalities in (1) and imply, in the limit as $s\to 1^{-}$, the Sobolev–Zhang
inequality.

######  Theorem 1.

For $0<s<1$ and $f\in W^{s,1}(\mathbb{R}^{n})$,

(4)
$\|f\|_{{\frac{n}{n-s}}}\leq\alpha_{n,s}n\omega_{n}^{({n+s})/n}\Big{(}\frac{1}{n}\int_{{\mathbb{S}}^{n-1}}\Big{(}\int_{0}^{\infty}t^{-s}\,\Big{\|}\frac{f(\cdot+t\xi)-f(\cdot)}{t}\Big{\|}_{1}\,\mathrm{d}t\Big{)}^{-n/s}\,\mathrm{d}\xi\Big{)}^{-s/n}\\\
\leq\alpha_{n,s}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{|x-y|^{n+s}}\,\mathrm{d}x\,\mathrm{d}y.$

There is equality in the first inequality if and only if $f$ is a constant
multiple of the indicator function of an ellipsoid. There is equality in the
second inequality if $f$ is radially symmetric.

In order to prove Theorem 1 we introduce the $s$-fractional polar projection
body $\operatorname{\Pi^{\ast,{\it s}}}\\!f$ associated to $f$, defined as the
star body whose gauge function for $\xi\in{\mathbb{S}}^{n-1}$ is

(5) $\|{\xi}\|_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}^{s}=\int_{0}^{\infty}t^{-s}\,\int_{\mathbb{R}^{n}}\Big{|}\frac{f(x+t\xi)-f(x)}{t}\Big{|}\,\mathrm{d}x\,\mathrm{d}t$

(see Section 3 for details). The affine fractional Sobolev inequality now can
be written as

(6)
$\|f\|_{{\frac{n}{n-s}}}\leq\alpha_{n,s}n\omega_{n}^{({n+s})/n}|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{-s/n},$

where $|\cdot|$ denotes $n$-dimensional Lebesgue measure. Since both sides of
(6) are invariant under translations of $f$, and for volume-preserving linear
transformations $\phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$,

$\operatorname{\Pi^{\ast,{\it
s}}}\\!\,(f\circ\phi^{-1})=\phi\operatorname{\Pi^{\ast,{\it s}}}\\!f,$

it follows that (6) is an affine inequality.

To show that Theorem 1 implies the Sobolev–Zhang inequality, we establish a
result corresponding to (2). For $f\in W^{1,1}(\mathbb{R}^{n})$, we will prove
that

(7) $\lim_{s\to 1^{-}}(1-s)|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{-s/n}=2\,|\operatorname{\Pi}^{*}\\!f|^{-1/n},$

where $\operatorname{\Pi}^{*}\\!f$ is the polar projection body of $f$ which
is defined (with different notation) in [38] (also see [24]) for
$\xi\in{\mathbb{S}}^{n-1}$ by

$\|\xi\|_{\operatorname{\Pi}^{*}\\!f}=\frac{1}{2}\int_{\mathbb{R}^{n}}|\langle\nabla
f(x),\xi\rangle|\,\mathrm{d}x.$

Moreover, we establish the corresponding statement for $f\in
BV(\mathbb{R}^{n})$ and also recover Wang’s generalized Sobolev–Zhang
inequality (without characterization of the equality case) from Theorem 1 (see
Section 4).

We prove Theorem 1 by first establishing affine fractional Pólya–Szegő
inequalities as a consequence of the Riesz rearrangement inequality. For the
equality case, we use anisotropic fractional Sobolev norms (introduced in
[21]) and the equality case of the Riesz rearrangement inequality due to
Burchard [5]. For $0<s<1$ and $E\subset\mathbb{R}^{n}$ of finite $s$-perimeter
(see Section 2), we define the fractional polar projection body as
$\operatorname{\Pi^{\ast,{\it s}}}\\!E=\operatorname{\Pi^{\ast,{\it
s}}}\\!\operatorname{1}_{E}$, where $\operatorname{1}_{E}$ is the indicator
function of $E$, and obtain from the affine fractional Pólya–Szegő
inequalities that

(8) $\Big{(}\frac{|\operatorname{\Pi^{\ast,{\it
s}}}\\!E|}{|\operatorname{\Pi^{\ast,{\it
s}}}\\!{B^{n}}|}\Big{)}^{-s/n}\geq\Big{(}\frac{|E|}{|{B^{n}}|}\Big{)}^{(n-s)/n}$

for $0<s<1$ and $E\subset\mathbb{R}^{n}$ of finite $s$-perimeter, where
${B^{n}}$ is the Euclidean unit ball in $\mathbb{R}^{n}$. From these
fractional Petty projection inequalities, Theorem 1 follows by a co-area
formula for the fractional Sobolev norm. In the limit as $s\to 1^{-}$, we
obtain from (8) the generalized Petty projection inequality by Zhang [38] and
Wang [35]. In addition, we prove versions of our functional and geometric
inequalities for Steiner symmetrization.

In Section 9, we connect fractional polar projection bodies to radial mean
bodies (introduced by Gardner and Zhang [13]) and obtain fractional Zhang
inequalities. Moreover, we establish new affine isoperimetric inequalities for
radial mean bodies, thereby proving a natural conjecture in the field.

##  2\. Preliminaries

###  2.1. Function spaces

For $p\geq 1$ and measurable $f:\mathbb{R}^{n}\to\mathbb{R}$, let

$\|f\|_{p}=\Big{(}\int_{\mathbb{R}^{n}}|f(x)|^{p}\,\mathrm{d}x\Big{)}^{1/p}.$

We set $\\{f\geq t\\}=\\{x\in\mathbb{R}^{n}:f(x)\geq t\\}$ for
$t\in\mathbb{R}$ and use corresponding notation for level sets, etc. We say
that $f$ is non-zero, if $\\{f\neq 0\\}$ has positive measure, and we identify
functions that are equal up to a set of measure zero. For $p\geq 1$, let

$L^{p}(\mathbb{R}^{n})=\Big{\\{}f:\mathbb{R}^{n}\to\mathbb{R}:f\text{ is
measurable},\|f\|_{p}<\infty\Big{\\}}.$

Here and below, when we use measurability and related notions, we refer to the
$n$-dimensional Lebesgue measure on $\mathbb{R}^{n}$.

For $E\subset\mathbb{R}^{n}$, the indicator function $\operatorname{1}_{E}$ is
defined by $\operatorname{1}_{E}(x)=1$ for $x\in E$ and
$\operatorname{1}_{E}(x)=0$ otherwise. We say that two measurable sets are
equivalent if their indicator functions are equal up to a set of measure zero.

Let $0<s<1$. We define the $s$-fractional Sobolev space as

$W^{s,1}(\mathbb{R}^{n})=\Big{\\{}f\in
L^{1}(\mathbb{R}^{n}):\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{|x-y|^{n+s}}\,\mathrm{d}x\,\mathrm{d}y<\infty\Big{\\}}.$

For $f\in L^{1}(\mathbb{R}^{n})$, we say that $f$ admits a weak gradient
$\nabla f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ if

$\int_{\mathbb{R}^{n}}\langle\nu(x),\nabla
f(x)\rangle\,\mathrm{d}x=-\int_{\mathbb{R}^{n}}f(x)\operatorname{div}\nu(x)\,\,\mathrm{d}x$

for every $\nu\in C^{\infty}_{c}(\mathbb{R}^{n};\mathbb{R}^{n})$, where
$C_{c}^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n})$ is the set of smooth vector
fields $\nu:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with compact support and
$\operatorname{div}\nu$ denotes the divergence of $\nu$. We set

$W^{1,1}(\mathbb{R}^{n})=\big{\\{}f\in L^{1}(\mathbb{R}^{n}):|\nabla f|\in
L^{1}(\mathbb{R}^{n})\big{\\}}.$

Let $\mathcal{B}(\mathbb{R}^{n})$ denote the class of Borel sets in
$\mathbb{R}^{n}$. For $f\in L^{1}(\mathbb{R}^{n})$, we say that $f$ is a
function of bounded variation on $\mathbb{R}^{n}$ if there is a finite vector-
valued Radon measure
$\mathrm{D}f:\mathcal{B}(\mathbb{R}^{n})\to\mathbb{R}^{n}$ such that

(9)
$\int_{\mathbb{R}^{n}}\langle\nu(x),\sigma_{f}(x)\rangle\,\mathrm{d}|\mathrm{D}f|(x)=-\int_{\mathbb{R}^{n}}f(x)\operatorname{div}\nu(x)\,\,\mathrm{d}x$

for every $\nu\in C_{c}^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n})$, where
$|\mathrm{D}f|:\mathcal{B}(\mathbb{R}^{n})\to[0,\infty)$ denotes the variation
measure of $\mathrm{D}f$ and $\sigma_{f}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ the
Radon–Nikodym derivative of $\mathrm{D}f$ with respect to $|\mathrm{D}f|$. We
write $BV(\mathbb{R}^{n})$ for the space of $L^{1}$ functions of bounded
variation. For more information on functions of bounded variation, we refer to
[8, Chapter 5].

###  2.2. Symmetrization

Let $E\subseteq\mathbb{R}^{n}$ be a Borel set of finite measure. The Schwarz
symmetral of $E$, denoted by $E^{\star}$, is the centered Euclidean ball with
same volume as $E$.

For a non-negative integrable function $f$ and $t>0$, the superlevel set
$\\{f\geq t\\}$ is a measurable set of finite measure. The layer cake formula
states that

$f(x)=\int_{0}^{\infty}\operatorname{1}_{\\{f\geq t\\}}(x)\,\mathrm{d}t$

for almost every $x\in\mathbb{R}^{n}$ and allows us to recover the function
from its superlevel sets. The Schwarz symmetral of $f$, denoted by
$f^{\star}$, is defined by

$f^{\star}(x)=\int_{0}^{\infty}\operatorname{1}_{\\{f\geq
t\\}^{\star}}(x)\,\mathrm{d}t$

for $x\in\mathbb{R}^{n}$. Hence $f^{\star}$ is determined by the properties of
being radially symmetric and having superlevel sets that are balls of the same
measure as the superlevel sets of $f$. Note that $f^{\star}$ is often called
symmetric decreasing rearrangement of $f$.

Let $\xi\in S^{n-1}$. The Steiner symmetral of $E$ in the direction $\xi$,
denoted by $E^{\xi}$, is defined by the property that for every line $L$
parallel to $\xi$, the set $L\cap E^{\xi}$ is an interval of the same length
as $L\cap E$ and symmetric with respect to the subspace orthogonal to $\xi$.

For a non-negative integrable function $f$, the Steiner symmetral $f^{\xi}$ is
defined by

$f^{\xi}(x)=\int_{0}^{\infty}\operatorname{1}_{\\{f\geq
t\\}^{\xi}}(x)\,\mathrm{d}t$

for $x\in\mathbb{R}^{n}$.

###  2.3. Star bodies

For a set $K\subseteq\mathbb{R}^{n}$ that is star-shaped (with respect to the
origin), the gauge function $\|\cdot\|_{K}:\mathbb{R}^{n}\to[0,\infty]$ is
defined as

$\|x\|_{K}=\inf\\{\lambda>0:x\in\lambda K\\},$

and the radial function $\rho_{K}:\mathbb{R}^{n}\setminus\\{0\\}\to[0,\infty]$
as

$\rho_{K}(x)=\|x\|_{K}^{-1}=\sup\\{\lambda\geq 0:\lambda x\in K\\}.$

We call $K$ a star body if its radial function is strictly positive and
continuous in $\mathbb{R}^{n}\setminus\\{0\\}$.

We say that a star-shaped set is $p$-convex with $0<p\leq 1$ if

$\|x+y\|_{K}^{p}\leq\|x\|_{K}^{p}+\|y\|_{K}^{p}$

for all $x,y\in\mathbb{R}^{n}$. A $p$-convex body is a star body that is also
$p$-convex. For more information on $p$-convex sets, see, for example, [16].

For star bodies $K,L\subseteq\mathbb{R}^{n}$ and $p\in\mathbb{R}$, define the
dual mixed volume as

$\tilde{V}_{p}(K,L)=\frac{1}{n}\int_{{\mathbb{S}}^{n-1}}\rho_{K}(\xi)^{n-p}\rho_{L}(\xi)^{p}\,\mathrm{d}\xi.$

Notice that

$\tilde{V}_{p}(K,K)=|K|.$

The dual mixed volume inequality (see [31, Section 9.3] or [12, B.29]) states
that for $p<0$ or $p>n$,

(10) $\tilde{V}_{p}(K,L)\geq|K|^{({n-p})/n}|L|^{p/n},$

and the reverse inequality holds for $0<p<n$. Equality holds for $p\neq 0,n$
if and only if $K$ and $L$ are dilates, where we say that star bodies $K$ and
$L$ are dilates if $\rho_{K}=c\,\rho_{L}$ on ${\mathbb{S}}^{n-1}$ for some
$c>0$.

###  2.4. Convex bodies and the Petty projection inequality

A set $K\subseteq\mathbb{R}^{n}$ is called a (proper) convex body if it is
compact and convex and has non-empty interior. For a convex body $K$, the
support function $h_{K}:\mathbb{R}^{n}\to\mathbb{R}$ is defined as

$h_{K}(y)=\max\\{\langle x,y\rangle:x\in K\\}.$

A convex body $K$ that contains the origin in its interior is also a star
body. The polar body of such $K$ is defined as

$K^{*}=\\{y\in\mathbb{R}^{n}:\langle x,y\rangle\leq 1\text{ for all }x\in
K\\}$

and is a convex body containing the origin in its interior while

$\|x\|_{K}=h_{K^{*}}(x)$

for $x\in\mathbb{R}^{n}$.

For $K\subset\mathbb{R}^{n}$ a convex body, the projection body is defined as
the convex body whose support function for $\xi\in{\mathbb{S}}^{n-1}$ is

(11)
$h_{\operatorname{\Pi}K}(\xi)=|\operatorname{proj}_{\xi^{\perp}}(K)|_{n-1},$

where $\operatorname{proj}_{\xi^{\perp}}$ denotes the orthogonal projection to
the hyperplane, $\xi^{\perp}$, orthogonal to $\xi$ and $|\cdot|_{n-1}$ stands
for $(n-1)$-dimensional volume in $\xi^{\perp}$. Using the surface area
measure $S(K,\cdot)$ of $K$, we can also write

(12)
$h_{\operatorname{\Pi}K}(\xi)=\frac{1}{2}\int_{{\mathbb{S}}^{n-1}}|\langle\xi,\eta\rangle|\,\mathrm{d}S(K,\eta)$

for $\xi\in{\mathbb{S}}^{n-1}$. Projection bodies were introduced by Minkowski
and $h_{\operatorname{\Pi}K}(\xi)$ is called the brightness of $K$ in the
direction $\xi$ (see [12, 31] for more information).

The important Petty projection inequality [28] states that for a convex body
$K\subset\mathbb{R}^{n}$,

(13)
$|K|^{(n-1)/n}\leq\frac{\omega_{n}}{\omega_{n-1}}|\operatorname{\Pi}^{*}\\!K|^{-1/n}$

with equality precisely for ellipsoids, where $\operatorname{\Pi}^{*}\\!K$ is
the polar body of $\operatorname{\Pi}K$. A new proof using Theorem 1 (and
hence the Riesz rearrangement inequality) is given in Section 7. A previous
proof of (13), that also uses the Riesz rearrangement inequality and the so-
called convolution square of a convex body, was given by Schmuckenschläger
[32]. For generalization of projection bodies of convex bodies, see, for
example, [2, 6, 18, 14, 23, 26, 20], and for further results related to the
Sobolev–Zhang inequality, see, for example, [15, 25, 24, 27, 36].

###  2.5. Anisotropic fractional Sobolev norms and perimeters

Let $0<s<1$ and let $K\subset\mathbb{R}^{n}$ be a star body. For $f\in
W^{s,1}(\mathbb{R}^{n})$, the anisotropic fractional Sobolev norm of $f$ with
respect to $K$ is

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{\|x-y\|_{K}^{n+s}}\,\mathrm{d}x\,\mathrm{d}y.$

It was introduced in [21] for $K$ a convex body (also, see [22]). The
$s$-fractional perimeter with respect to $K$ of a measurable set
$E\subseteq\mathbb{R}^{n}$ was introduced in [21] as

$P_{s}(E,K)=\int_{E}\int_{E^{c}}\frac{1}{\|x-y\|_{K}^{n+s}}\,\mathrm{d}x\,\mathrm{d}y,$

where $E^{c}=\mathbb{R}^{n}\backslash E$ is the complement of $E$ in
$\mathbb{R}^{n}$. The same definition will be used for
$K\subset\mathbb{R}^{n}$ a star body. The anisotropic $s$-fractional perimeter
of $E$ with respect to $K$ is half of the $s$-fractional anisotropic Sobolev
norm of $\operatorname{1}_{E}$ with respect to $K$. Since $K$ is a star body,
this implies for measurable $E\subset\mathbb{R}^{n}$ that $P_{s}(E,K)<\infty$
if and only if $\operatorname{1}_{E}\in W^{s,1}(\mathbb{R}^{n})$. If $K$ is
the Euclidean unit ball, we write $P_{s}(E)$ and obtain the well-known
Euclidean $s$-fractional perimeter of $E$. We will use that

(14)
$P_{s}({B^{n}})=\frac{2^{1-s}\pi^{(n-1)/2}n\omega_{n}}{s(n-s)}\frac{\Gamma\left(\frac{1-s}{2}\right)}{\Gamma\left(\frac{n-s}{2}\right)}=\frac{2^{1-s}n\omega_{n}\omega_{n-s}}{s(1-s)\omega_{1-s}}$

(cf. [9]), where $\omega_{q}=\pi^{q/2}/\Gamma(q/2+1)$ for $q>0$ and $\Gamma$
is the gamma function. A simple way to obtain this result is by using the
Blaschke–Petkantschin formula [33, Theorem 7.2.7]:

(15) $\begin{split}P_{s}({B^{n}})&=\int_{{B^{n}}\cap
L\neq\emptyset}\int_{{B^{n}}\cap L}\int_{(\mathbb{R}^{n}\backslash{B^{n}})\cap
L}\frac{\,\mathrm{d}H^{1}(x)\,\mathrm{d}H^{1}(y)}{|x-y|^{s+1}}\,\mathrm{d}L\\\
&=\frac{1}{2}\int_{{\mathbb{S}}^{n-1}}\int_{\xi^{\perp}}\int_{{B^{n}}\cap(z+\mathbb{R}\xi)}\int_{(\mathbb{R}^{n}\backslash{B^{n}})\cap(z+\mathbb{R}\xi)}\frac{\,\mathrm{d}H^{1}(x)\,\mathrm{d}H^{1}(y)}{|x-y|^{1+s}}\,\mathrm{d}z\,\mathrm{d}\xi\\\
&=\frac{n\omega_{n}(n-1)\omega_{n-1}}{2}\int_{0}^{1}\int_{|x|\leq\sqrt{1-t^{2}}}\int_{|y|\geq\sqrt{1-t^{2}}}\frac{\,\mathrm{d}H^{1}(x)\,\mathrm{d}H^{1}(y)}{|x-y|^{1+s}}\,t^{n-2}\,\mathrm{d}t\\\
&=\frac{2^{1-s}(n-1)n\omega_{n-1}\omega_{n}}{s(1-s)}\int_{0}^{1}(1-t^{2})^{(1-s)/2}t^{n-2}\,\mathrm{d}t\\\
&=\frac{(n-1)n\omega_{n-1}\omega_{n}}{2^{s}s(1-s)}\,{\rm
B}(\frac{1-s}{2}+1,\frac{n-1}{2})\\\ \end{split}$

where the outer integration in the first integral is on the affine
Grassmannian of lines in $\mathbb{R}^{n}$ and we use the $(n-1)$-dimensional
Hausdorff measure for the direction of lines and the $(n-1)$-dimensional
Lebesgue measure in the orthogonal complement of a line. Here $H^{1}$ denotes
$1$-dimensional Hausdorff measure and $\mathbb{R}\xi$ is the line with
direction $\xi$ while $\rm B$ is the beta function. For the optimal constant
in (1), we get

(16) $\alpha_{n,s}=2P_{s}({B^{n}})\,\omega_{n}^{-(n-s)/n}$

since there is equality in (1) for indicator functions of balls.

The following fractional co-area formula is a consequence of Fubini’s theorem
(cf. [34, Example (2.9)]):

(17)
$\begin{split}\,\,\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{\|x-y\|_{K}^{n+s}}\,\mathrm{d}x\,\mathrm{d}y&=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\frac{|\operatorname{1}_{\\{f\geq
t\\}}(x)-\operatorname{1}_{\\{f\geq
t\\}}(y)|}{\|x-y\|_{K}^{n+s}}\,\mathrm{d}t\,\mathrm{d}x\,\mathrm{d}y\\\
&=2\int_{0}^{\infty}P_{s}(\\{f\geq t\\},K)\,\mathrm{d}t\end{split}$

for non-negative $f\in W^{s,1}(\mathbb{R}^{n})$ and a star body
$K\subset\mathbb{R}^{n}$.

##  3\. Fractional Polar Projection Bodies

Let $0<s<1$ and $f\in BV(\mathbb{R}^{n})$. We discuss properties of the
$s$-fractional polar projection body $\operatorname{\Pi^{\ast,{\it s}}}\\!f$,
that is, the star-shaped set defined through its gauge function for
$\xi\in\mathbb{R}^{n}$ as

(18) $\|{\xi}\|_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}^{s}=\int_{0}^{\infty}t^{-s}\,\int_{\mathbb{R}^{n}}\Big{|}\frac{f(x+t\xi)-f(x)}{t}\Big{|}\,\mathrm{d}x\,\mathrm{d}t.$

Notice that $\|\cdot\|_{\operatorname{\Pi^{\ast,{\it s}}}\\!f}$ is a
$1$-homogeneous function.

We start with a simple observation. Let $K\subset\mathbb{R}^{n}$ be a star
body. Using (18), we obtain

(19)
$\begin{split}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}&\frac{|f(x)-f(y)|}{\|x-y\|_{K}^{n+s}}\,\mathrm{d}x\,\mathrm{d}y\\\
&=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(y+z)-f(y)|}{\|z\|_{K}^{n+s}}\,\mathrm{d}z\,\mathrm{d}y\\\
&=\int_{{\mathbb{S}}^{n-1}}\int_{0}^{\infty}{\|t\xi\|_{K}^{-n-s}}\int_{\mathbb{R}^{n}}{|f(y+t\xi)-f(y)|}\,t^{n-1}\,\mathrm{d}y\,\mathrm{d}t\,\mathrm{d}\xi\\\
&=\int_{{\mathbb{S}}^{n-1}}{\|\xi\|_{K}^{-n-s}}\int_{0}^{\infty}t^{-s}\Big{\|}\frac{f(\cdot+t\xi)-f(\cdot)}{t}\Big{\|}_{1}\,\mathrm{d}t\,\mathrm{d}\xi\\\
&=\int_{{\mathbb{S}}^{n-1}}\rho_{K}(\xi)^{n+s}\rho_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}(\xi)^{-s}\,\mathrm{d}\xi.\end{split}$

Hence

(20)
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{\|x-y\|_{K}^{n+s}}\,\mathrm{d}x\,\mathrm{d}y=n\,\tilde{V}_{-s}(K,\operatorname{\Pi^{\ast,{\it
s}}}\\!f),$

where we have already used the following result, which shows that
$\operatorname{\Pi^{\ast,{\it s}}}\\!f$ is an origin-symmetric star body.

######  Proposition 2.

For non-zero $f\in W^{s,1}(\mathbb{R}^{n})$, the set
$\,\operatorname{\Pi^{\ast,{\it s}}}\\!f$ is an origin-symmetric $s$-convex
body with the origin in its interior. Moreover, there is $c>0$ depending only
on $f$ such that $\operatorname{\Pi^{\ast,{\it s}}}\\!f\subseteq c\,{B^{n}}$
for every $s\in(0,1)$.

###### Proof.

First, note that since for $\xi\in\mathbb{R}^{n}$ and $t>0$,

$\int_{\mathbb{R}^{n}}|{f(x-t\xi)-f(x)}|\,\mathrm{d}x=\int_{\mathbb{R}^{n}}|f(x)-f(x+t\xi)|\,\mathrm{d}x,$

the set $\operatorname{\Pi^{\ast,{\it s}}}\\!f$ is origin-symmetric.

Next, we show that $\operatorname{\Pi^{\ast,{\it s}}}\\!f$ is bounded. We take
$r>1$ large enough so that
$\int_{r{B^{n}}}|f(x)|\,\mathrm{d}x\geq\frac{2}{3}\|f\|_{1}$ and easily see
that for $t>r$,

$\displaystyle\int_{\mathbb{R}^{n}}|f(x+t\xi)-$ $\displaystyle
f(x)|\,\mathrm{d}x$
$\displaystyle\geq\int_{r{B^{n}}}(|f(x)|-|f(x+t\xi)|)\,\mathrm{d}x+\int_{r{B^{n}}-t\xi}(|f(x+t\xi)|-|f(x)|)\,\mathrm{d}x$
$\displaystyle\geq 2(\frac{2}{3}\|f\|_{1}-\frac{1}{3}\|f\|_{1}).$

Hence,

$\displaystyle\int_{0}^{\infty}t^{-s}\left\|\frac{f(x+t\xi)-f(x)}{t}\right\|_{1}\,\mathrm{d}t\geq\frac{2}{3}\|f\|_{1}\int_{r}^{\infty}t^{-s-1}\,\mathrm{d}t\geq\frac{2}{3r}\|f\|_{1},$

which implies that $\operatorname{\Pi^{\ast,{\it s}}}\\!f\subseteq c\,{B^{n}}$
for $c>0$ independent of $s$.

Next, we show that $\|\cdot\|_{\operatorname{\Pi^{\ast,{\it s}}}\\!f}^{s}$ is
sublinear on $\mathbb{R}^{n}$. Indeed, for $\xi,\eta\in\mathbb{R}^{n}$, the
triangle inequality and a change of variables show that

(21) $\displaystyle\|\xi+\eta\|_{\operatorname{\Pi^{\ast,{\it s}}}\\!f}^{s}$
$\displaystyle=\int_{0}^{\infty}t^{-s-1}\|f(\cdot+t\xi+t\eta)-f(\cdot)\|_{1}\,\mathrm{d}t$
(22)
$\displaystyle\leq\int_{0}^{\infty}t^{-s-1}(\|f(\cdot+t\xi+t\eta)-f(\cdot+t\xi)\|_{1}+\|f(\cdot+t\xi)-f(\cdot)\|_{1})\,\mathrm{d}t$
(23) $\displaystyle=\|\xi\|_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}^{s}+\|\eta\|_{\operatorname{\Pi^{\ast,{\it s}}}\\!f}^{s},$

which shows that $\operatorname{\Pi^{\ast,{\it s}}}\\!f$ is an $s$-convex set.

Now, we show that $\operatorname{\Pi^{\ast,{\it s}}}\\!f$ has the origin in
its interior. Using the relation (20), we get

(24) $\int_{{\mathbb{S}}^{n-1}}\|\xi\|_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}^{s}\,\mathrm{d}\xi=\frac{1}{n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{|x-y|^{n+s}}\,\mathrm{d}x\,\mathrm{d}y,$

which is finite since $f\in W^{s,1}(\mathbb{R}^{n})$. We choose $r>0$ large
enough so that the set $A=\\{\xi\in
S^{n-1}:\|\xi\|_{\operatorname{\Pi^{\ast,{\it s}}}\\!f}^{s}<r\\}$ has positive
$(n-1)$-dimensional Hausdorff measure and a basis
$\\{\xi_{1},\ldots,\xi_{n}\\}\subseteq A$ of $\mathbb{R}^{n}$. Applying (if
necessary) a linear transformation to $\operatorname{\Pi^{\ast,{\it s}}}\\!f$
, we may assume without loss of generality that $\xi_{i}=e_{i}$ are the
canonical basis vectors. For every $x\in\mathbb{R}^{n}$ writing $x=\sum
x_{i}e_{i}$ and using (22) we get

$\|x\|_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}\leq\left(\sum|x_{i}|^{s}\|e_{i}\|_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}^{s}\right)^{1/s}\leq d\,|x|,$

where $d>0$ is independent of $x$. This shows that
$\operatorname{\Pi^{\ast,{\it s}}}\\!f$ has the origin as interior point.

Finally, it is easy to see that a star-shaped set that is $s$-convex and has
the origin in its interior, also has a continuous gauge function, being thus
an $s$-convex body. ∎

##  4\. The Limit Case $s\to 1^{-}$ of Fractional Polar Projection Bodies

We prove (7) and the corresponding statement for functions of bounded
variation. For $f\in BV(\mathbb{R}^{n})$, the polar projection body is defined
for $\xi\in{\mathbb{S}}^{n-1}$ by

(25)
$\|\xi\|_{\operatorname{\Pi}^{*}\\!f}=\frac{1}{2}\int_{\mathbb{R}^{n}}|\langle\sigma_{f}(x),\xi\rangle|\,\mathrm{d}|\mathrm{D}f|(x)$

(see [35]). Note that for $f\in W^{1,1}(\mathbb{R}^{n})$, this definition
coincides with (5).

######  Theorem 3.

Let $f\in BV(\mathbb{R}^{n})$. For $\xi\in{\mathbb{S}}^{n-1}$,

$\lim_{s\to 1^{-}}(1-s)\|\xi\|_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}^{s}=2\,\|\xi\|_{\operatorname{\Pi}^{*}\\!f}.$

Moreover,

$\lim_{s\to 1^{-}}(1-s)|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{-s/n}=2\,|\operatorname{\Pi}^{*}\\!f|^{-1/n}$

and

(26) $\lim_{s\to 1^{-}}(1-s)\tilde{V}_{-s}(K,\operatorname{\Pi^{\ast,{\it
s}}}\\!f)=2\,\tilde{V}_{-1}(K,\operatorname{\Pi}^{*}\\!f)$

for every star body $K\subset\mathbb{R}^{n}$.

We require the following two lemmas.

######  Lemma 4.

If $\varphi:[0,\infty)\to[0,\infty)$ is a measurable function with $\lim_{t\to
0^{+}}\varphi(t)=\varphi(0)$ and such that
$\int_{0}^{\infty}t^{-s_{0}}\varphi(t)\,\mathrm{d}t<\infty$ for some
$s_{0}\in(0,1)$, then

$\lim_{s\to
1^{-}}(1-s)\int_{0}^{\infty}t^{-s}\varphi(t)\,\mathrm{d}t=\varphi(0).$

###### Proof.

Given $\varepsilon>0$ take $\delta\in(0,1)$ so that $0\leq t\leq\delta$
implies $|\varphi(t)-\varphi(0)|\leq\varepsilon$. We have

(27) $\displaystyle\int_{0}^{\infty}t^{-s}\varphi(t)\,\mathrm{d}t$
$\displaystyle=\int_{0}^{\delta}t^{-s}\varphi(0)\,\mathrm{d}t+\int_{0}^{\delta}t^{-s}(\varphi(t)-\varphi(0))\,\mathrm{d}t+\int_{\delta}^{\infty}t^{-s}\varphi(t)\,\mathrm{d}t$
(28) $\displaystyle=\frac{\delta^{1-s}}{1-s}\varphi(0)+A+B$

where $(1-s)|A|\leq\varepsilon\delta^{1-s}\leq\varepsilon$ and

$|B|\leq\delta^{s_{0}-s}\int_{\delta}^{\infty}t^{-s_{0}}\varphi(t)\,\mathrm{d}t\leq\delta^{s_{0}-s}\int_{0}^{\infty}t^{-s_{0}}\varphi(t)\,\mathrm{d}t$

for $s\in(s_{0},1)$. Since $(1-s)B\to 0$ and $\delta^{1-s}\to 1$ as $s\to
1^{-}$, we can take $s$ sufficiently close to $1$ so that
$(1-s)|B|\leq\varepsilon$ and
$|\delta^{1-s}\varphi(0)-\varphi(0)|\leq\varepsilon$, obtaining
$\left|(1-s)\int_{0}^{\infty}t^{-s}\varphi(t)\,\mathrm{d}t-\varphi(0)\right|\leq
3\varepsilon$. ∎

######  Lemma 5.

For $f\in BV(\mathbb{R}^{n})$ and $\xi\in{\mathbb{S}}^{n-1}$,

$\lim_{t\to
0}\Big{\|}\frac{f(\cdot+t\xi)-f(\cdot)}{t}\Big{\|}_{1}=\int_{\mathbb{R}^{n}}|\langle\sigma_{f}(x),\xi\rangle|\,\mathrm{d}|\mathrm{D}f|(x).$

###### Proof.

Let $g:\mathbb{R}^{n}\to\mathbb{R}$ be a smooth function with compact support,
and write $\operatorname{div}_{x}$ for the divergence taken with respect to
the variable $x$. Using (9), we obtain

$\displaystyle\int_{\mathbb{R}^{n}}g(x)\,\frac{f(x+t\xi)-f(x)}{t}\,\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{n}}f(x)\,\frac{g(x-t\xi)-g(x)}{t}\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{n}}f(x)\int_{0}^{1}\langle\nabla
g(x-rt\xi),\xi\rangle\,\mathrm{d}r\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{n}}f(x)\operatorname{div}_{x}\Big{(}\int_{0}^{1}g(x-rt\xi)\,\mathrm{d}r\,\xi\Big{)}\,\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{1}g(x-rt\xi)dr\Big{)}\langle\sigma_{f}(x),\xi\rangle\,\mathrm{d}|Df|(x).$

Therefore

(29) $\displaystyle\lim_{t\to
0}\int_{\mathbb{R}^{n}}g(x)\,\frac{f(x+t\xi)-f(x)}{t}\,\mathrm{d}x=\int_{\mathbb{R}^{n}}g(x)\langle\sigma_{f}(x),\xi\rangle\,\mathrm{d}|Df|(x).$

Let $\varepsilon>0$. Since $|\mathrm{D}f|$ is a finite Radon measure, the set
of smooth functions with compact support is dense in the space of $L^{1}$
functions w.r.t. $|\mathrm{D}f|$ (cf. [10, Proposition 7.9]). Hence, there is
a smooth function $g:\mathbb{R}^{n}\to\mathbb{R}$ with compact support such
that

$\int_{\mathbb{R}^{n}}|g(x)-\operatorname{sgn}(\langle\sigma_{f}(x),\xi\rangle)|\,\mathrm{d}|\mathrm{D}f|(x)<\varepsilon$

and $\|g\|_{\infty}\leq 1+\varepsilon$. By (29),

(30) $\displaystyle(1+\varepsilon)\liminf_{t\to 0}$
$\displaystyle\int_{\mathbb{R}^{n}}\Big{|}\frac{f(x+t\xi)-f(x)}{t}\Big{|}\,\mathrm{d}x$
(31) $\displaystyle\geq\lim_{t\to
0}\int_{\mathbb{R}^{n}}g(x)\frac{f(x+t\xi)-f(x)}{t}\,\mathrm{d}x$ (32)
$\displaystyle=\int_{\mathbb{R}^{n}}g(x)\langle\sigma_{f}(x),\xi\rangle\,\mathrm{d}|\mathrm{D}f|(x)$
(33)
$\displaystyle=\int_{\mathbb{R}^{n}}|\langle\sigma_{f}(x),\xi\rangle|\,\mathrm{d}|\mathrm{D}f|(x)$
(34)
$\displaystyle\phantom{=}\,+\int_{\mathbb{R}^{n}}(g(x)-\operatorname{sgn}(\langle\sigma_{f}(x),\xi\rangle))\langle\sigma_{f}(x),\xi\rangle\,\mathrm{d}|\mathrm{D}f|(x)$
(35)
$\displaystyle\geq\int_{\mathbb{R}^{n}}|\langle\sigma_{f}(x),\xi\rangle|\,\mathrm{d}|\mathrm{D}f|(x)-\varepsilon.$

Since $\varepsilon>0$ was arbitrary, we obtain

$\liminf_{t\to
0}\Big{\|}\frac{f(\cdot+t\xi)-f(\cdot)}{t}\Big{\|}_{1}\geq\int_{\mathbb{R}^{n}}|\langle\sigma_{f}(x),\xi\rangle|\,\mathrm{d}|\mathrm{D}f|(x).$

For the opposite inequality, notice that

$\left|\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{1}h(x-rt\xi)\,\mathrm{d}r\Big{)}\langle\sigma_{f}(x),\xi\rangle\,\mathrm{d}|\mathrm{D}f|(x)\right|\leq\|h\|_{\infty}\int_{\mathbb{R}^{n}}|\langle\sigma_{f}(x),\xi\rangle|\,\mathrm{d}|\mathrm{D}f|(x)$

for every $h\in L^{\infty}(\mathbb{R}^{n})$. ∎

###### Proof of Theorem 3.

Define $\varphi:[0,\infty)\to[0,\infty)$ by

$\varphi(t)=\Big{\|}\frac{f(\cdot+t\xi)-f(\cdot)}{t}\Big{\|}_{1}$

and note that $\varphi(t)\leq\frac{2\|f\|_{1}}{t}$ for $t>0$. By Lemma 4 and
Lemma 5,

$\lim_{s\to
1^{-}}(1-s)\int_{0}^{\infty}t^{-s}\Big{\|}\frac{f(\cdot+t\xi)-f(\cdot)}{t}\Big{\|}_{1}\,\mathrm{d}t=\int_{\mathbb{R}^{n}}|\langle\sigma_{f}(x),\xi\rangle|\,\mathrm{d}|\mathrm{D}f|(x).$

By Proposition 2 we can use the dominated convergence theorem to obtain,

$\displaystyle\lim_{s\to 1^{-}}n\,$
$\displaystyle|(1-s)^{-1/s}\operatorname{\Pi^{\ast,{\it s}}}\\!f|$
$\displaystyle=\lim_{s\to
1^{-}}\int_{{\mathbb{S}}^{n-1}}\Big{(}(1-s)\int_{0}^{\infty}t^{-s}\Big{\|}\frac{f(\cdot+t\xi)-f(\cdot)}{t}\Big{\|}_{1}\,\mathrm{d}t\Big{)}^{-n/s}\,\mathrm{d}\xi$
$\displaystyle=\int_{{\mathbb{S}}^{n-1}}\Big{(}\int_{\mathbb{R}^{n}}|\langle\sigma_{f}(x),\xi\rangle|\,\mathrm{d}|\mathrm{D}f|(x)\Big{)}^{-n}\,\mathrm{d}\xi$
$\displaystyle=n\,|\tfrac{1}{2}\operatorname{\Pi}^{*}\\!f|$

and

$\displaystyle\lim_{s\to
1^{-}}n(1-s)\tilde{V}_{-s}(K,\operatorname{\Pi^{\ast,{\it s}}}\\!f)$
$\displaystyle=\lim_{s\to
1^{-}}(1-s)\int_{{\mathbb{S}}^{n-1}}\|\xi\|_{K}^{n+s}\|\xi\|_{\operatorname{\Pi^{\ast,{\it
s}}}\\!f}^{s}\,\mathrm{d}\xi$
$\displaystyle=2\int_{{\mathbb{S}}^{n-1}}\|\xi\|_{K}^{n}\,\|\xi\|_{\operatorname{\Pi}^{*}\\!f}\,\mathrm{d}\xi$
$\displaystyle=2n\,\tilde{V}_{-1}(K,\operatorname{\Pi}^{*}\\!f),$

which completes the proof of the theorem. ∎

##  5\. Anisotropic Fractional Pólya–Szegő Inequalities

Almgren and Lieb [1] established the following Pólya–Szegő inequality for
fractional Sobolev norms. The equality case was settled by Frank and Seiringer
[11]. Let $0<s<1$ and $n\geq 1$. For non-negative $f\in
W^{s,1}(\mathbb{R}^{n})$,

(36)
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{|x-y|^{n+s}}\,\mathrm{d}x\,\mathrm{d}y\geq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f^{\star}(x)-f^{\star}(y)|}{|x-y|^{n+s}}\,\mathrm{d}x\,\mathrm{d}y,$

with equality precisely if $\\{f\geq t\\}$ is equivalent to a ball for almost
every $t>0$.

The following improvement of (36) is key to prove the main theorems in Section
8. It is a variation of [17, Theorem 3.1].

######  Theorem 6.

Let $0<s<1$ and $K\subset\mathbb{R}^{n}$ a star body. For
$f:\mathbb{R}^{n}\to\mathbb{R}$ a non-negative integrable function,

(37)
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{\|x-y\|^{n+s}_{K}}\,\mathrm{d}x\,\mathrm{d}y\geq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f^{\star}(x)-f^{\star}(y)|}{\|x-y\|^{n+s}_{K^{\star}}}\,\mathrm{d}x\,\mathrm{d}y.$

If equality holds for non-zero $f\in W^{s,1}(\mathbb{R}^{n})$, then $K$ is a
centered ellipsoid and for almost every $t>0$, the level set $\\{f\geq t\\}$
is equivalent to an ellipsoid.

The inequality in Theorem 6 follows rather directly from the Riesz
rearrangement inequality, which is stated in full generality, for example, in
[4].

######  Theorem 7 (Riesz’s rearrangement inequality).

For $f,g,k:\mathbb{R}^{n}\to\mathbb{R}$ non-negative measurable functions,

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}f(x)k(x-y)g(y)\,\mathrm{d}x\,\mathrm{d}y\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}f^{\star}(x)k^{\star}(x-y)g^{\star}(y)\,\mathrm{d}x\,\mathrm{d}y.$

To establish the equality case in Theorem 6, we use the characterization of
equality cases in a version of the Riesz rearrangement inequality due to
Burchard [5].

######  Theorem 8 (Burchard).

Let $A,B$ and $C$ be sets of finite positive measure in $\,\mathbb{R}^{n}$ and
denote by $\alpha,\beta$ and $\gamma$ the radii of their Schwarz symmetrals
$A^{\star},B^{\star}$ and $C^{\star}$. For
$\,|\alpha-\beta|<\gamma<\alpha+\beta$, there is equality in

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\operatorname{1}_{A}(y)\operatorname{1}_{B}(x-y)\operatorname{1}_{C}(x)\,\mathrm{d}x\,\mathrm{d}y\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\operatorname{1}_{A^{\star}}(y)\operatorname{1}_{B^{\star}}(x-y)\operatorname{1}_{C^{\star}}(x)\,\mathrm{d}x\,\mathrm{d}y$

if and only if, up to sets of measure zero,

$A=a+\alpha D,B=b+\beta D,C=c+\gamma D,$

where $D$ is a centered ellipsoid, and $a,b$ and $c=a+b$ are vectors in
$\mathbb{R}^{n}$.

By the co-area formula (17), Theorem 6 is an immediate consequence of the
following result for anisotropic fractional perimeters.

######  Theorem 9.

Let $0<s<1$ and $K\subset\mathbb{R}^{n}$ a star body. For
$E\subset\mathbb{R}^{n}$ measurable,

(38)
$\int_{E}\int_{E^{c}}\frac{1}{\|x-y\|^{n+s}_{K}}\,\mathrm{d}x\,\mathrm{d}y\geq\int_{E^{\star}}\int_{(E^{\star})^{c}}\frac{1}{\|x-y\|^{n+s}_{K^{\star}}}\,\mathrm{d}x\,\mathrm{d}y.$

If equality holds for $E$ with finite positive $s$-perimeter, then $K$ is a
centered ellipsoid and $E$ is equivalent to an ellipsoid.

###### Proof.

Write

$\|z\|_{K}^{-n-s}=\int_{0}^{\infty}k_{t}(z)\,\mathrm{d}t\text{, where
}k_{t}(z)=\operatorname{1}_{t^{-1/({n+s})}K}(z)$

and use Fubini’s theorem, to obtain

$\int_{E}\int_{E^{c}}\frac{1}{\|x-y\|^{n+s}_{K}}\,\mathrm{d}x\,\mathrm{d}y=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\operatorname{1}_{E}(x)\operatorname{1}_{E^{c}}(y)k_{t}(x-y)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}t.$

For fixed $t\in(0,\infty)$, we have

(39)
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\operatorname{1}_{E}(x)$
$\displaystyle\operatorname{1}_{E^{c}}(y)k_{t}(x-y)\,\mathrm{d}x\,\mathrm{d}y$
(40)
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\big{(}\operatorname{1}_{E}(x)-\operatorname{1}_{E}(x)\operatorname{1}_{E}(y)\big{)}k_{t}(x-y)\,\mathrm{d}x\,\mathrm{d}y$
(41)
$\displaystyle=t^{-\frac{n}{n+s}}|K||E|-\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\operatorname{1}_{E}(x)\,k_{t}(x-y)\operatorname{1}_{E}(y)\,\mathrm{d}x\,\mathrm{d}y.$

Clearly the first term is invariant under Schwarz symmetrization, and for the
second term we can apply Theorem 7 to show that the left side does not
increase under Schwarz symmetrization. Thus

(42)
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\operatorname{1}_{E}(x)k_{t}(x-y)\operatorname{1}_{E^{c}}(y)\,\mathrm{d}x\,\mathrm{d}y\geq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\operatorname{1}_{E^{\star}}(x)\,k^{\star}_{t}(x-y)\operatorname{1}_{(E^{\star})^{c}}(y)\,\mathrm{d}x\,\mathrm{d}y$

for $t>0$ and integrating this inequality for $t\in(0,\infty)$, we obtain
(38).

If there is equality in (38), it follows from (40) that for almost every
$t>0$,

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\operatorname{1}_{E}(x)\operatorname{1}_{t^{-1/(n+s)}K}(x-y)\operatorname{1}_{E}(y)\,\mathrm{d}x\,\mathrm{d}y$

is invariant under Schwarz symmetrization. Now we may apply Theorem 8 with
$A=C=E$ and $B=t^{-1/({n+s})}K$ for suitable $t>0$. Since $A=C$ we have $a=c$
and this implies $b=0$. Hence $K$ is a centered ellipsoid and $E$ is
equivalent to an ellipsoid. ∎

To establish corresponding inequalities for Steiner symmetrization, we require
the following result.

######  Theorem 10 (Rogers [29]).

Let $\xi\in{\mathbb{S}}^{n-1}$. For $f,g,k:\mathbb{R}^{n}\to\mathbb{R}$ non-
negative and measurable,

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}f(x)k(x-y)g(y)\,\mathrm{d}x\,\mathrm{d}y\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}f^{\xi}(x)k^{\xi}(x-y)g^{\xi}(y)\,\mathrm{d}x\,\mathrm{d}y.$

We remark that the above theorem is a special case of the inequalities of
Rogers [30] and Brascamp–Lieb–Luttinger [4, Lemma 3.2].

As in the case of Schwarz symmetrization, we obtain the following
consequences.

######  Theorem 11.

Let $0<s<1$ and $\xi\in{\mathbb{S}}^{n-1}$. If $K\subset\mathbb{R}^{n}$ is a
star body, then

(43)
$\int_{E}\int_{E^{c}}\frac{1}{\|x-y\|^{n+s}_{K}}\,\mathrm{d}x\,\mathrm{d}y\geq\int_{E^{\xi}}\int_{(E^{\xi})^{c}}\frac{1}{\|x-y\|^{n+s}_{K^{\xi}}}\,\mathrm{d}x\,\mathrm{d}y$

for $E\subset\mathbb{R}^{n}$ measurable.

######  Theorem 12.

Let $0<s<1$ and $\xi\in{\mathbb{S}}^{n-1}$. If $K\subset\mathbb{R}^{n}$ is a
star body, then

(44)
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{\|x-y\|^{n+s}_{K}}\,\mathrm{d}x\,\mathrm{d}y\geq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f^{\xi}(x)-f^{\xi}(y)|}{\|x-y\|^{n+s}_{K^{\xi}}}\,\mathrm{d}x\,\mathrm{d}y$

for $f:\mathbb{R}^{n}\to\mathbb{R}$ non-negative and integrable.

##  6\. Affine Fractional Pólya–Szegő Inequalities

We establish the following affine version of the fractional Pólya–Szegő
inequality (36) by Almgren and Lieb.

######  Theorem 13.

For $0<s<1$ and non-negative $f\in W^{s,1}(\mathbb{R}^{n})$,

(45) $|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{-s/n}\geq|\operatorname{\Pi^{\ast,{\it s}}}\\!f^{\star}|^{-s/n},$

with equality precisely if $\\{f\geq t\\}$ is equivalent to an ellipsoid for
almost every $t>0$.

###### Proof.

By Theorem 6, (20) and (10), we have

(46) $\displaystyle\tilde{V}_{-s}(K,\operatorname{\Pi^{\ast,{\it s}}}\\!f)$
$\displaystyle\geq\tilde{V}_{-s}(K^{\star},\operatorname{\Pi^{\ast,{\it
s}}}\\!f^{\star})$ (47)
$\displaystyle\geq|K^{\star}|^{({n+s})/{n}}|\operatorname{\Pi^{\ast,{\it
s}}}\\!f^{\star}|^{-s/n}$ (48)
$\displaystyle=|K|^{({n+s})/{n}}|\operatorname{\Pi^{\ast,{\it
s}}}\\!f^{\star}|^{-s/n}.$

Setting $K=\operatorname{\Pi^{\ast,{\it s}}}\\!f$, we obtain that

(49) $\displaystyle|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|=\tilde{V}_{-s}(\operatorname{\Pi^{\ast,{\it
s}}}\\!f,\operatorname{\Pi^{\ast,{\it
s}}}\\!f)\geq|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{({n+s})/{n}}|\operatorname{\Pi^{\ast,{\it
s}}}\\!f^{\star}|^{-\frac{s}{n}},$

which completes the proof of the inequality. The equality case follows from
the equality case of the dual mixed volumes inequality and of Theorem 6. ∎

If we multiply (45) by $(1-s)$ and let $s\to 1^{-}$, we obtain from Theorem 13
and Theorem 3 the following affine Pólya–Szegő inequality by Cianchi, Lutwak,
Yang and Zhang [6] (see also [27]):

(50)
$|\operatorname{\Pi}^{*}\\!f|^{-1/n}\geq|\operatorname{\Pi}^{*}\\!f^{\star}|^{-1/n}$

for $f\in W^{1,1}(\mathbb{R}^{n})$.

Using the same proof as for Theorem 13 for Steiner symmetrization and
replacing Theorem 7 by Theorem 12, we obtain the following result.

######  Theorem 14.

Let $0<s<1$ and $\xi\in{\mathbb{S}}^{n-1}$. Then

(51) $|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{-s/n}\geq|\operatorname{\Pi^{\ast,{\it s}}}\\!f^{\xi}|^{-s/n}$

for non-negative $f\in W^{s,1}(\mathbb{R}^{n})$.

As before, we obtain from Theorem 14 and Theorem 3 the following affine
inequality:

(52)
$|\operatorname{\Pi}^{*}\\!f|^{-1/n}\geq|\operatorname{\Pi}^{*}\\!f^{\xi}|^{-1/n}$

for $f\in W^{1,1}(\mathbb{R}^{n})$ and $\xi\in{\mathbb{S}}^{n-1}$.

##  7\. Fractional Petty Projection Inequalities

A set $E\subset\mathbb{R}^{n}$ of finite measure is a set of finite perimeter
if its indicator function $\operatorname{1}_{E}$ is in $BV(\mathbb{R}^{n})$.
This allows us to translate results for functions in $BV(\mathbb{R}^{n})$ to
sets of finite perimeter and finite measure. The following result is an
immediate consequence of the affine fractional Pólya–Szegő inequality from
Theorem 13.

######  Theorem 15.

For $\,0<s<1$ and $E\subset\mathbb{R}^{n}$ of finite $s$-perimeter and finite
measure,

$|\operatorname{\Pi^{\ast,{\it
s}}}\\!E|^{-s/n}\geq|\operatorname{\Pi^{\ast,{\it s}}}\\!E^{\star}|^{-s/n}$

with equality precisely if $E$ is equivalent to an ellipsoid.

Here we write $\operatorname{\Pi^{\ast,{\it s}}}\\!E$ for the fractional polar
projection body of $\operatorname{1}_{E}$. Note that

$\|\xi\|^{s}_{\operatorname{\Pi^{\ast,{\it
s}}}\\!E}=\int_{0}^{\infty}t^{-s}\frac{|E\triangle(E-t\xi)|}{t}\,\mathrm{d}t$

for $E\subset\mathbb{R}^{n}$ of finite $s$-perimeter and finite measure. Also
note that (20) implies that

(53)
$2\int_{E}\int_{E^{c}}\frac{1}{\|x-y\|_{K}^{n+s}}\,\mathrm{d}x\,\mathrm{d}y=n\,\tilde{V}_{-s}(K,\operatorname{\Pi^{\ast,{\it
s}}}\\!E)$

for $K\subset\mathbb{R}^{n}$ a star body.

From Theorem 15, we easily obtain the first inequality of the following
result, which we call the fractional Petty projection inequality.

######  Theorem 16.

For $\,0<s<1$ and $E\subset\mathbb{R}^{n}$ of finite $s$-perimeter and finite
measure,

(54)
$\Big{(}\frac{|E|}{|{B^{n}}|}\Big{)}^{{(n-s)}/n}\leq\Big{(}\frac{|\operatorname{\Pi^{\ast,{\it
s}}}\\!E|}{|\operatorname{\Pi^{\ast,{\it s}}}\\!{B^{n}}|}\Big{)}^{-s/n}\\\
\leq\frac{P_{s}(E)}{P_{s}({B^{n}})}.$

There is equality in the first inequality if and only if $E$ is equivalent to
an ellipsoid. There is equality in the second inequality if and only if $E$ is
equivalent to a set of constant $s$-fractional brightness.

Here, we say that $E\subset\mathbb{R}^{n}$ is of constant $s$-fractional
brightness if $\operatorname{\Pi^{\ast,{\it s}}}\\!E$ is a dilate of
${B^{n}}$. The second inequality in (54) and its equality case follow directly
from (20) and the dual mixed volume inequality (10).

Let $E\subset\mathbb{R}^{n}$ be a set of finite perimeter and finite measure.
The perimeter of $E$ is defined as $P(E)=|\mathrm{D}\operatorname{1}_{E}|$. By
Dávila’s version of (2),

$\lim_{s\to 1^{-}}\frac{P_{s}(E)}{P_{s}({B^{n}})}=\frac{P(E)}{P({B^{n}})}.$

Taking the limit $s\to 1^{-}$ in (54), we obtain from this and Theorem 3 that

(55)
$\Big{(}\frac{|E|}{|{B^{n}}|}\Big{)}^{{(n-1)}/n}\leq\Big{(}\frac{|\operatorname{\Pi}^{*}\\!E|}{|\operatorname{\Pi}^{*}\\!{B^{n}}|}\Big{)}^{-1/n}\\\
\leq\frac{P(E)}{P({B^{n}})}$

for $E\subset\mathbb{R}^{n}$ of finite perimeter and finite measure. The first
inequality is the generalized Petty projection inequality that was proved by
Gaoyong Zhang [38] for compact sets with piecewise $C^{1}$ boundary and in
full generality by Tuo Wang [35]. For $E$ a convex body, it is the classical
Petty projection inequality (13). There is equality in the second inequality
of (55) if and only if $E$ is of constant brightness.

The Steiner inequality for fractional polar projection bodies is contained in
the following result, which is an immediate consequence of Theorem 14.

######  Theorem 17.

Let $\,0<s<1$ and $\xi\in{\mathbb{S}}^{n-1}$. Then

(56) $|\operatorname{\Pi^{\ast,{\it
s}}}\\!E|^{-s/n}\geq|\operatorname{\Pi^{\ast,{\it s}}}\\!E^{\xi}|^{-s/n}$

for $E\subset\mathbb{R}^{n}$ of finite $s$-perimeter and finite Lebesgue
measure.

Taking the limit as $s\to 1^{-}$ and using Theorem 3, we obtain from Theorem
17 the following Steiner inequality for polar projection bodies, which was
recently established by Youjiang Lin [19]:

$|\operatorname{\Pi}^{*}\\!E|^{-1/n}\geq|\operatorname{\Pi}^{*}\\!E^{\xi}|^{-1/n}$

for $E\subset\mathbb{R}^{n}$ of finite perimeter and finite Lebesgue measure
and $\xi\in{\mathbb{S}}^{n-1}$.

##  8\. Proof of Theorem 1

A version of the following lemma for $s=1$ is contained in [38].

######  Lemma 18.

Let $0<s<1$. If $g:\mathbb{R}^{n}\to[0,\infty)$ is measurable, then

(57)
$\displaystyle\Big{(}\int_{\mathbb{R}^{n}}g(x)^{n/({n-s})}\,\mathrm{d}x\Big{)}^{({n-s})/n}$
$\displaystyle\leq\int_{0}^{\infty}|\\{g\geq t\\}|^{({n-s})/n}\,\mathrm{d}t.$

If the right side is finite, then there is equality precisely if
$g=c\operatorname{1}_{E}$ for some $E\subset\mathbb{R}^{n}$ of finite measure
and $c\geq 0$.

###### Proof.

Let the right side of (57) be finite. By Fubini’s theorem, we have

(58) $\displaystyle\int_{\mathbb{R}^{n}}g(x)^{n/({n-s})}\,\mathrm{d}x$
$\displaystyle=\frac{n}{n-s}\int_{0}^{\infty}t^{s/({n-s})}|\\{g\geq
t\\}|\,\mathrm{d}t.$

Since $r\mapsto|\\{g\geq r\\}|$ is monotone decreasing, we obtain for $t>0$,

(59) $\displaystyle t^{s/({n-s})}|\\{g\geq t\\}|$
$\displaystyle=\big{(}t\,|\\{g\geq
t\\}|^{({n-s})/n}\big{)}^{s/({n-s})}|\\{g\geq t\\}|^{({n-s})/n}$ (60)
$\displaystyle\leq\big{(}\int_{0}^{t}|\\{g\geq
r\\}|^{({n-s})/n}\,\mathrm{d}r\big{)}^{s/({n-s})}|\\{g\geq t\\}|^{({n-s})/n}$
(61)
$\displaystyle=\frac{n-s}{n}\frac{\,\mathrm{d}\,\,}{\,\mathrm{d}t}\big{(}\int_{0}^{t}|\\{g\geq
r\\}|^{({n-s})/n}\,\mathrm{d}r\big{)}^{n/({n-s})}.$

Hence

(62)
$\displaystyle\Big{(}\int_{\mathbb{R}^{n}}g(x)^{n/({n-s})}\,\mathrm{d}x\Big{)}^{({n-s})/n}$
$\displaystyle\leq\int_{0}^{\infty}|\\{g\geq t\\}|^{({n-s})/n}\,\mathrm{d}t$

and there is equality precisely if $g=c\operatorname{1}_{E}$ for some
$E\subset\mathbb{R}^{n}$ of finite measure and $c\geq 0$. ∎

###### Proof of Theorem 1.

First, assume that $f$ is non-negative. By the co-area formula (17) and (20),

(63)
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{\|x-y\|_{K}}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=2\int_{0}^{\infty}P_{s}(\\{f\geq t\\},K)\,\mathrm{d}t$ (64)
$\displaystyle=n\int_{0}^{\infty}\tilde{V}_{-s}(K,\operatorname{\Pi^{\ast,{\it
s}}}\\!\,\\{f\geq t\\})\,\mathrm{d}t.$

Hence, by the dual mixed volume inequality (10) and (20),

(65) $\displaystyle|\operatorname{\Pi^{\ast,{\it s}}}\\!f|$
$\displaystyle=\int_{0}^{\infty}\tilde{V}_{-s}(\operatorname{\Pi^{\ast,{\it
s}}}\\!f,\operatorname{\Pi^{\ast,{\it s}}}\\!\,\\{f\geq t\\})\,\mathrm{d}t$
(66) $\displaystyle\geq|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{({n+s})/n}\int_{0}^{\infty}|\operatorname{\Pi^{\ast,{\it
s}}}\\!\,\\{f\geq t\\}|^{-s/n}\,\mathrm{d}t$

and

(67) $|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{-s/n}\geq\int_{0}^{\infty}|\operatorname{\Pi^{\ast,{\it
s}}}\\!\,\\{f\geq t\\}|^{-s/n}\,\mathrm{d}t.$

The equality cases of inequality (1) and of the dual mixed volume inequality,
applied to (53) for $K={B^{n}}$, show that

${\alpha_{n,s}n\omega_{n}^{({n+s})/n}}=|\operatorname{\Pi^{\ast,{\it
s}}}\\!{B^{n}}|^{s/n}{|{B^{n}}|^{({n-s})/n}}.$

By the fractional Petty projection inequality from Theorem 16, it now follows
that

(68) $\displaystyle|\operatorname{\Pi^{\ast,{\it s}}}\\!f|^{-s/n}$
$\displaystyle\geq\frac{|\operatorname{\Pi^{\ast,{\it
s}}}\\!{B^{n}}|^{-s/n}}{|{B^{n}}|^{({n-s})/n}}\int_{0}^{\infty}|\operatorname{\Pi^{\ast,{\it
s}}}\\!\,\\{f\geq t\\}|^{-s/n}\,\mathrm{d}t$ (69)
$\displaystyle\geq\frac{1}{\alpha_{n,s}n\omega_{n}^{({n+s})/n}}\int_{0}^{\infty}|\\{f\geq
t\\}|^{(n-s)/s}\,\mathrm{d}t.$

In particular, the last term is finite. We apply Lemma 18 to obtain

(70)
$\displaystyle\alpha_{n,s}n\omega_{n}^{(n+s)/n}|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{-s/n}\geq\|f\|_{\frac{n}{n-s}}.$

Combining the equality cases of the fractional Petty projection inequality and
Lemma 18, we obtain the equality case for non-negative $f$.

For general $f$ and $x,y\in\mathbb{R}^{n}$, we use
$|f(x)-f(y)|\geq||f(x)|-|f(y)||$, where equality holds if and only if $f(x)$
and $f(y)$ are both non-negative or non-positive. Applying this inequality in
the definition of $\operatorname{\Pi^{\ast,{\it s}}}\\!f$ we obtain

$|\operatorname{\Pi^{\ast,{\it
s}}}\\!{|f|}|^{-s/n}\leq|\operatorname{\Pi^{\ast,{\it s}}}\\!{f}|^{-s/n}$

with equality if and only if $f$ has constant sign for almost every
$x,y\in\mathbb{R}^{n}$. Using (70) for $|f|$, we obtain the first inequality
of the theorem and its equality case.

For the second inequality, we set $K={B^{n}}$ in (20) and apply the dual mixed
volume inequality (10) to obtain

(71)
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|}{|x-y|^{n+s}}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=n\tilde{V}_{-s}({B^{n}},\operatorname{\Pi^{\ast,{\it
s}}}\\!f)\geq n\omega_{n}^{({n+s})/n}|\operatorname{\Pi^{\ast,{\it
s}}}\\!f|^{-s/n}.$

There is equality precisely if $\operatorname{\Pi^{\ast,{\it s}}}\\!f$ is a
ball, which is the case for radially symmetric functions. ∎

##  9\. Fractional Zhang Projection Inequalities and Radial Mean Bodies

Let $E\subset\mathbb{R}^{n}$ be a convex body. For $p>-1$, Gardner and Zhang
[13] defined the radial $p$-th mean body of $E$, by its radial function for
$\xi\in{\mathbb{S}}^{n-1}$, as

$\rho_{\operatorname{R}_{p}\\!E}(\xi)^{p}=\frac{1}{|E|}\int_{E}\rho_{E-x}(\xi)^{p}\,\mathrm{d}x$

for $p\neq 0$ and as

$\log(\rho_{\operatorname{R}_{0}\\!E}(\xi))=\frac{1}{|E|}\int_{E}\log(\rho_{E-x}(\xi))\,\mathrm{d}x.$

They showed that $\operatorname{R}_{p}\\!E$ is a star body for $p>-1$ and a
convex body for $p\geq 0$.

Let $0<s<1$. For a star body $K\subset\mathbb{R}^{n}$ and a convex body
$E\subset\mathbb{R}^{n}$, we obtain from Fubini’s theorem that

(72)
$\displaystyle\int_{E}\int_{E^{c}}\frac{1}{\|x-y\|_{K}^{n+s}}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=\int_{E}\int_{(E-x)^{c}}\frac{1}{\|z\|_{K}^{n+s}}\,\mathrm{d}z\,\mathrm{d}x$
(73)
$\displaystyle=\frac{1}{s}\int_{E}\int_{{\mathbb{S}}^{n-1}}\rho_{K}(\xi)^{n+s}\rho_{E-x}(\xi)^{-s}\,\mathrm{d}\xi\,\mathrm{d}x$
(74)
$\displaystyle=\frac{|E|}{s}\int_{{\mathbb{S}}^{n-1}}\rho_{K}(\xi)^{n+s}\rho_{\operatorname{R}_{-s}\\!E}(\xi)^{-s}\,\mathrm{d}\xi.$

Hence, by (53),

(75) $\tilde{V}_{-s}(K,\operatorname{\Pi^{\ast,{\it
s}}}\\!E)=\frac{2|E|}{s}\,\tilde{V}_{-s}(K,\operatorname{R}_{-s}\\!E)$

for every star body $K$. By the equality case of the dual mixed volume
inequality (10), it follows that

(76) $\operatorname{\Pi^{\ast,{\it
s}}}\\!E=\Big{(}\frac{s}{2|E|}\Big{)}^{\frac{1}{s}}\operatorname{R}_{-s}\\!E$

for every convex body $E\subset\mathbb{R}^{n}$.

Hence, we can reformulate inequalities obtained by Gardner and Zhang [13] to
obtain inequalities for fractional polar projection bodies. In particular,
Theorem 5.5 and Lemma 5.7 from [13] imply that

(77) $\Big{(}\frac{|\operatorname{\Pi^{\ast,{\it
s}}}\\!E|}{|\operatorname{\Pi^{\ast,{\it
s}}}\\!\Delta|}\Big{)}^{-s/n}\leq\Big{(}\frac{|E|}{|\Delta|}\Big{)}^{{(n-s)}/n}$

for $E\subset\mathbb{R}^{n}$ a convex body and $0<s<1$ with equality precisely
if $E$ is a simplex, where $\Delta$ is any $n$-dimensional simplex in
$\mathbb{R}^{n}$. Letting $s\to 1^{-}$ and using Theorem 3, we obtain the
Zhang projection inequality [37] (without the equality case), which was
reproved in [13]:

(78)
$\Big{(}\frac{|\operatorname{\Pi}^{*}\\!E|}{|\operatorname{\Pi}^{*}\\!\Delta|}\Big{)}^{-1/n}\leq\Big{(}\frac{|E|}{|\Delta|}\Big{)}^{{(n-1)}/n}$

for $E\subset\mathbb{R}^{n}$ a convex body.

Conversely, we can reformulate Theorem 16 for radial mean bodies and obtain
sharp affine isoperimetric inequalities for radial $p$-th mean bodies for
$-1<p<0$. Using also a variation of our approach to Theorem 16, we obtain the
following result, which proves a natural conjecture in the field. We have
included the case $p=n$ from [13, Lemma 5.7] in the statement of the result.

######  Theorem 19.

If $\,E\subset\mathbb{R}^{n}$ is a convex body, then

(79)
$\begin{array}[]{rcll}\displaystyle\frac{|\operatorname{R}_{p}\\!E|}{|E|}&\geq&\displaystyle\frac{|\operatorname{R}_{p}\\!{B^{n}}|}{|{B^{n}}|}&\quad\text{
for }\,-1<p<0\,\text{ and }\,p>n,\\\\[12.0pt]
\displaystyle\frac{|\operatorname{R}_{p}\\!E|}{|E|}&\leq&\displaystyle\frac{|\operatorname{R}_{p}\\!{B^{n}}|}{|{B^{n}}|}&\quad\text{
for }\,0<p<n,\end{array}$

with equality if and only if $E$ is an ellipsoid, where for $p>-1$ and $p\neq
0,n$,

(80)
$\frac{|\operatorname{R}_{p}\\!{B^{n}}|}{|{B^{n}}|}=\displaystyle\Big{(}\frac{2^{p+1}\omega_{n+p}}{(p+1)\omega_{n}\omega_{p+1}}\Big{)}^{n/p}.$

Moreover,

$\frac{|\operatorname{R}_{0}\\!E|}{|E|}=2^{n}e^{\frac{n}{2}(\psi(\frac{1}{2})-\psi(\frac{n}{2}+1))}\quad\text{
and }\quad\frac{|\operatorname{R}_{n}\\!E|}{|E|}=1$

for every convex body $E\subset\mathbb{R}^{n}$, where $\psi$ is the digamma
function.

###### Proof.

Let $K\subset\mathbb{R}^{n}$ be a star body. It follows from Fubini’s theorem
that

(81)
$\begin{split}\int_{E}\int_{E}\frac{1}{\|x-y\|_{K}^{n-p}}\,\mathrm{d}x\,\mathrm{d}y&=\int_{E}\int_{E-x}\frac{1}{\|z\|_{K}^{n-p}}\,\mathrm{d}z\,\mathrm{d}x\\\
&=\frac{1}{p}\int_{E}\int_{{\mathbb{S}}^{n-1}}\rho_{K}(\xi)^{n-p}\rho_{E-x}(\xi)^{p}\,\mathrm{d}\xi\,\mathrm{d}x\\\
&=\frac{|E|}{p}\int_{{\mathbb{S}}^{n-1}}\rho_{K}(\xi)^{n-p}\rho_{\operatorname{R}_{p}\\!E}(\xi)^{p}\,\mathrm{d}\xi\\\
&=\frac{n|E|}{p}\,\tilde{V}_{p}(K,\operatorname{R}_{p}\\!E).\end{split}$

For $0<p<n$, the Riesz rearrangement inequality, Theorem 7, implies that

(82) $\displaystyle\int_{E}\int_{E}\frac{1}{\|x-y\|_{K}^{n-p}}$
$\displaystyle\,\mathrm{d}x\,\mathrm{d}y$ (83)
$\displaystyle\leq\int_{E^{\star}}\int_{E^{\star}}\frac{1}{\|x-y\|_{K^{\star}}^{n-p}}\,\mathrm{d}x\,\mathrm{d}y$
(84)
$\displaystyle=\Big{(}\frac{|E|}{|{B^{n}}|}\Big{)}^{(n+p)/n}\Big{(}\frac{|K|}{|{B^{n}}|}\Big{)}^{(n-p)/n}\int_{{B^{n}}}\int_{{B^{n}}}\frac{1}{|x-y|^{n-p}}\,\mathrm{d}x\,\mathrm{d}y.$

As in the proof of Theorem 9, we obtain from Theorem 8 that there is equality
precisely if $E$ is an ellipsoid. Hence, setting $K=\operatorname{R}_{p}\\!E$,
we obtain from (81) that

(85)
$\displaystyle\frac{|\operatorname{R}_{p}\\!E|}{|\operatorname{R}_{p}\\!{B^{n}}|}$
$\displaystyle=\frac{p}{n|E||\operatorname{R}_{p}\\!{B^{n}}|}\,\int_{E}\int_{E}\frac{1}{\|x-y\|_{\operatorname{R}_{p}\\!E}^{n-p}}\,\mathrm{d}x\,\mathrm{d}y$
(86)
$\displaystyle\leq\Big{(}\frac{|E|}{|{B^{n}}|}\Big{)}^{p/n}\Big{(}\frac{|\operatorname{R}_{p}\\!E|}{|\operatorname{R}_{p}\\!{B^{n}}|}\Big{)}^{(n-p)/n}$

and the result follows.

For $p>n$, we choose $c>0$ so large that $c-\|x-y\|_{K}^{p-n}$ is non-negative
on $E\times E$. This is possible, since $E$ is compact. We apply the Riesz
rearrangement inequality and its equality case and obtain the result.

Next, we calculate the constants for $p>0$ and $-1<p<0$. For $p>0$, we obtain
by (81) that

(87)
$\displaystyle\Big{(}\frac{|\operatorname{R}_{p}\\!{B^{n}}|}{|{B^{n}}|}\Big{)}^{1/n}$
$\displaystyle=\Big{(}\frac{p}{n\omega_{n}^{2}}\int_{B^{n}}\int_{{B^{n}}}\frac{1}{|x-y|^{n-p}}\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{1/p}$

and by [33, Theorem 7.2.7 and Theorem 8.6.6] that

$\int_{B^{n}}\int_{{B^{n}}}\frac{1}{|x-y|^{n-p}}\,\mathrm{d}x\,\mathrm{d}y=\frac{2^{p+1}n\omega_{n}\omega_{n+p}}{p(p+1)\omega_{p+1}}.$

For $-1<p<0$, we obtain by (73) that

(88)
$\displaystyle\Big{(}\frac{|\operatorname{R}_{p}\\!{B^{n}}|}{|{B^{n}}|}\Big{)}^{1/n}$
$\displaystyle=\Big{(}\frac{-p}{n\omega_{n}^{2}}\int_{B^{n}}\int_{{B^{n}}^{c}}\frac{1}{|x-y|^{n-p}}\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{1/p}$

and obtain the result from (14).

The case $p=n$ follows from the polar coordinate formula of volume and the
definition of the radial $n$-th mean body. To obtain the statement for $p=0$,
we take the limit as $p\to 0$ in the second equation of (79), use that
$|\operatorname{R}_{p}\\!E|$ depends continuously on $p$ and obtain

(89)
$\frac{|\operatorname{R}_{0}\\!E|}{|E|}=\frac{|\operatorname{R}_{0}\\!{B^{n}}|}{|{B^{n}}|}$

for every convex body $E\subset\mathbb{R}^{n}$. By (81),

(90)
$\displaystyle\Big{(}\frac{|\operatorname{R}_{p}\\!{B^{n}}|}{|{B^{n}}|}\Big{)}^{1/n}$
$\displaystyle=\Big{(}\frac{p}{n\omega_{n}^{2}}\int_{B^{n}}\int_{{B^{n}}}\frac{1}{|x-y|^{n-p}}\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{1/p}$

for $p>0$. By [33, Theorem 7.2.7 and Theorem 8.6.6],

$\int_{B^{n}}\int_{{B^{n}}}\frac{1}{|x-y|^{n-p}}\,\mathrm{d}x\,\mathrm{d}y=\frac{2^{p+1}n\omega_{n}\omega_{n+p}}{p(p+1)\omega_{p+1}},$

where $\omega_{q}=\pi^{q/2}/\Gamma(q/2+1)$ for $q>0$. We obtain

(91)
$\displaystyle\Big{(}\frac{|\operatorname{R}_{0}\\!{B^{n}}|}{|{B^{n}}|}\Big{)}^{1/n}$
$\displaystyle=\lim_{p\to
0^{+}}\Big{(}\frac{2^{p+1}\omega_{n+p}}{(p+1)\omega_{n}\omega_{p+1}}\Big{)}^{1/p}={2\,e^{\frac{1}{2}(\psi(\frac{1}{2})-\psi(\frac{n}{2}+1))}},$

which concludes the proof. ∎

### Acknowledgments

J. Haddad was partially supported by CNPq, reference no. PQ 305559/2021-4 and
by PIADI2020, reference no. P20$\\_$00664\. M. Ludwig was supported, in part,
by the Austrian Science Fund (FWF): P 34446.

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