# Applying SDN to Mobile Networks: A New Perspective for 6G Architecture

Rashmi Yadav  Department of Electrical Engineering
Indian Institute of Technology Kanpur,
India
<EMAIL_ADDRESS>Rashmi Kamran  Department of Electrical Engineering,
Indian Institute of Technology Bombay,
India
<EMAIL_ADDRESS>Pranav Jha  Abhay Karandikar  Department of
Electrical Engineering,
Indian Institute of Technology Bombay,
India
<EMAIL_ADDRESS>Director, Indian Institute Technology Kanpur, India
<EMAIL_ADDRESS>
Department of Electrical Engineering,
Indian Institute of Technology Bombay, India
<EMAIL_ADDRESS>

###### Abstract

The upcoming Sixth Generation (6G) mobile communications system envisions
supporting a variety of use cases with differing characteristics, e.g., very
low to extremely high data rates, diverse latency needs, ultra massive
connectivity, sustainable communications, ultra-wide coverage etc. To
accommodate these diverse use cases, the 6G system architecture needs to be
scalable, modular, and flexible; both in its user plane and the control plane.
In this paper, we identify some limitations of the existing Fifth Generation
System (5GS) architecture, especially that of its control plane. Further, we
propose a novel architecture for the 6G System (6GS) employing Software
Defined Networking (SDN) technology to address these limitations of the
control plane. The control plane in existing 5GS supports two different
categories of functionalities – handling end user signalling (e.g., user
registration, authentication) and control of user plane functions. We propose
to move the “end-user signalling functionality” out of the mobile network
control plane and treat it as user service, i.e., as payload or data. This
proposal results in an evolved service-driven architecture for mobile networks
bringing increased simplicity, modularity, scalability, flexibility and
security to its control plane. The proposed architecture can also support
service specific signalling support, if needed, making it better suited for
diverse 6GS use cases. To demonstrate the advantages of the proposed
architecture, we also compare its performance with the 5GS using a process
algebra-based simulation tool.

###### Index Terms:

Software-defined networking, Mobile networks, Service-driven architecture.

##  I Introduction

The notable rise in the range of diverse use cases with differing attributes
has paved the way for the continued evolution of mobile networks. The upcoming
6th Generation Mobile Communication System (6GS) is envisioned to support peak
data rate ($\geq$200 Gbps), very high mobility (500-1000 Km/h), very low
latency (0.1-1 ms), connection density in the range of 106-108 devices/Km2,
reliability of 10-5-10-7 [1]. Moreover, it is expected to witness further
diversity of use cases with the emergence of newer categories of use cases.
Focus Group on Technologies for Network 2030 (FG NET-2030) [2] has identified
and included the following use cases in its report: Holographic-type
communications, Tactile Internet for Remote Operations, Intelligent Operation
Networks, Network and Computing Convergence, Digital Twin, Space-Terrestrial
Integrated Network, Industrial IoT with cloudification etc. A scalable,
flexible and modular network architecture is one of the essential ingredients
towards tackling this immense diversity of use cases in future mobile
networks. Third Generation Partnership Project (3GPP) adopted technologies
such as Network Function Virtualization, Control and User Plane Separation,
Network slicing for Fifth Generation System (5GS), which resulted in improved
scalability and flexibility of 5GS over the previous generation mobile
communications systems such as Fourth Generation System (4GS).

However, there is scope for further improvement in mobile network architecture
especially that of its control plane through the application of Software
Defined Networking (SDN) technology. A survey of the existing research related
to SDN-based enhancements in the mobile network control plane is presented
next. The work in [3] proposes a centralised control plane for multi-Radio
Access Technology (multi-RAT) Radio Access Network (RAN) to enhance the
simplicity and flexibility of the network. Relocation of the control plane
functionality of RAN to the Core Network (CN) to reduce the signalling cost
between RAN and core has been discussed in [4]. Authors in [5] proposed a
decentralized control plane architecture for the 5GS with independent control
functions for different control events for flexible and scalable networks. An
SDN architecture where a middle cell and a middle cell controller are
introduced between the macro cell and the small cell to reduce the control
overhead of the macro cell and to address the scalability problems is proposed
in [6]. In [7], authors proposed a new 5GS core architecture based on the SDN
concept. They introduced a centralised SDN controller for easier and more
flexible management of the user plane. In [8], a hierarchical control plane is
designed to lighten the load of the controller. It focuses on the vertical
scalability of the control plane. In [9], a scalability metric for the SDN
control plane is proposed. Besides, a comparison between different SDN
architectures is analysed via mathematical methods. In addition, there is a
vast amount of literature on SDN-based network architectures, albeit unrelated
to mobile networks [10], [11].

To summarize, current research in the context of the application of SDN
technology to mobile networks mainly focuses on the centralized or distributed
architecture of the control plane for reduced control overheads or scalability
purposes. However, to the best of our knowledge, there is a limited
discussion/rethink on certain other aspects of network architecture, such as,
what functionality should constitute the mobile network control plane within
an SDN-based framework. Is the network control plane right place for “end user
signalling handling” functionality? Should “Non-Access Stratum (NAS) messages”
be handled by CN control plane functions such as Access and Mobility
Management Function (AMF) or should this functionality be moved out of AMF?
Should the user authentication function (Authentication Server Function (AUSF)
in 5GS) be part of the CN control plane? These questions assume even more
importance in the upcoming 6GS era, where a massive increase in the number of
UEs is expected and an accompanying growth in end-user signalling has the
potential to over-burden the network control plane. In one of our earlier
works [12], we briefly analysed these questions.

In order to bring in additional enhancements to mobile network architecture,
especially to its control plane, we propose to separate end user (User
Equipment (UE)) signalling handling from the control plane functions. In a
significant departure from the existing cellular networks, the proposed
architecture views UE signalling as payload, i.e., a form of data traversing
through the cellular network, not much different from other types of data such
as Video streaming or Web browsing. We analyse the proposed architecture using
Performance Evaluation Process Algebra (PEPA) [13], a formal language used to
model distributed systems. We also provide a comparative analysis of the
proposed architecture and the existing 5GS architecture through example call
flows for Protocol Data Unit (PDU) session establishment and UE handover
procedures. We demonstrate a significant reduction in the number of control
messages exchanged in the proposed architecture along with the network’s
scalability.

The rest of the paper is organised as follows: Section II provides limitations
of the existing 5GS mobile network architecture. Section III provides an
overview of the proposed architecture and highlights its advantages. Section
IV includes an information flow comparison of the existing and proposed
architecture for PDU session establishment and handover procedures. Section V
describes the system model using PEPA. Section VI covers the performance
analysis. Section VII provides the conclusion and future work.

##  II Limitations of existing 5GS Architecture

In this section, we have captured some of the limitations of the existing 5GS
architecture especially that of its control plane. Although there can be other
limitations too say pertaining to radio technology, etc., those are not
discussed here.

###  II-A Tight coupling of user plane control and UE signalling in control
plane

The existing 5GS architecture supports the control and user plane separation.
The 5GS control plane performs user plane control (network resource control,
e.g., setting up data path through the user plane) and UE signalling handling
functionalities (e.g., NAS/RRC (Radio Resource Control) message exchange with
UEs). There is a tight coupling between these two categories of
functionalities, i.e., between user plane control and UE signalling handling
and certain CN (e.g., AMF) and RAN gNodeB-Centralized Unit-Control Plane (gNB-
CU-CP) control plane functions in the existing 5GS perform both. A detailed
description of control plane functionality is provided in [14]. As
demonstrated here, decoupling of UE signalling handling functionality from
User plane control functionality may lead to a more modular and scalable
network architecture.

###  II-B Limited alignment with SDN paradigm

SDN is a networking paradigm which separates the control plane of a network
from its user (data) plane and centralizes the network’s intelligence in the
control plane. Although there are differing views in industry/academia on how
to define an SDN-based network architecture, we can still discern a broad
agreement on the topic [5], [15], [16]. The existing 5GS architecture
incorporates the concept of SDN, resulting in architectural features such as
the separation of the user plane from the control plane [14]. However, closer
observation shows that the 5GS architecture does not align completely with the
SDN paradigm. Besides controlling the user plane, the 5GS control plane also
exchanges signalling messages with UEs to provide services such as
authentication and also collect service requirements, e.g., requirements for
PDU connectivity service. The functionality of signalling exchange with UEs
may fit better within the service plane instead of the control plane.

###  II-C Non-uniform handling of services

Services in the existing 5GS can be categorized into the following two types:

  1. 1.

Application-based services such as Media streaming services, IP Multimedia
subsystem services, Mission-critical services, Multicast/Broadcast Services
(MBS) etc.

  2. 2.

Other than these application-based services, the 5GS network also provides
services such as initial access, registration, authentication, PDU
connectivity (connectivity to data networks), and connected mode mobility
support. Such services can be called built-in (or intrinsic) network services.

The two categories of services (Application based services and built-in
network services) are enabled differently in the 5GS. As Application (Service)
Functions (AFs) are independent and decoupled from the core and RAN functions
of mobile networks, they access the control plane functions of the mobile CN
over a standardized interface to enable service delivery through the user
plane. However, the delivery of built-in services is tightly integrated within
the control plane of the 5GS network (RAN and CN) itself. It also leads to the
usage of special paths for signalling exchange with UEs, different from the
regular data paths and brings certain inconsistencies to the architecture. For
example, the Performance Measurement Function (PMF), a sub-function within the
User Plane Function (UPF), exchanges “Measurement Assistance Information”, a
type of signalling information with UEs to aid the access traffic steering,
switching, and splitting (ATSSS) functionality at UPF. This signalling
information is exchanged via a regular data path (i.e. user plane) between the
UE and the PMF. This mechanism is different from how other signalling
information such as “radio measurement reports” to support the handover
procedure is exchanged.

###  II-D Complex protocols between control plane and user plane

The existing 5GS control plane architecture impacts the interface design
(protocols) between the control and user planes. For instance, F1 Application
Protocol (F1AP) is the protocol used on the interface between the RAN control
plane (gNB-CU-CP) and the RAN user plane (gNB-Distributed Unit (gNB-DU) or
RAN-DU). It is used to configure gNB-DU and also carries RRC (UE signalling)
messages for UEs. Integrating both these types of functionalities in a single
protocol results in a relatively complex communication protocol between gNB-
CU-CP and gNB-DU.

Figure 1: Control plane architecture for proposed architecture [17]

##  III Service driven architecture for 6GS mobile networks

This section presents the proposed architecture, which addresses the
architectural limitations of the existing 5GS (as discussed in Section II) and
highlights a few other advantages. In the proposed work, we aim to separate
the UE signalling handling from the control plane and treat them as a service
to the user to enhance modularity and flexibility in the mobile network
control plane. With the proposed separation, the control plane is left with
only the user plane control functionality, as shown in Fig. 1. The UE
signalling handling functionality is moved out of the control plane to the
service/application plane. The service plane consists of various in-built and
external service functions, as shown in Fig. 1, such as the PDU Session
Service Function (handles PDU session establishment and management providing
PDU connectivity service), Mobility Service Function (responsible for handling
UE mobility), Registration Service Function (handles UE registration with the
network), Authentication Service Function (manages UE authentication),
Multicast/Broadcast Services and a few others. Due to the reorganisation of
the architecture, it offers various architectural and performance advantages
discussed next. Please note that there may be separate controllers in the CN
and RAN, as shown in Fig. 3. Similarly, we have a separate resource plane
(user plane) for RAN and the CN. Further, the proposed architecture’s user or
resource plane may remain the same as the 3GPP 5GS.

###  III-A Advantages of the proposed 6GS architecture

This section highlights a few advantages of the proposed work. Segregation of
UE signalling handling functionality from the control plane simplifies the
control plane, which enhances the modularity of the control plane. The
reorganised architecture also aligns well with the SDN paradigm as the control
plane is redesigned to perform only user plane control functionality as
discussed in Section II-B. The proposed architecture also allows internal (or
built-in 5GS) services to be treated the same way as external application-
based services, leading to uniform handling of various services. Further, this
proposal results in the simplification of the control messages. For instance,
the number of sessions management-related messages is reduced due to the setup
of a direct path between UE and the service function (detailed in Section
IV-B), leading to simplified call flows. Also, the number of hops between the
RAN controller and the CN controller in the proposed architecture is less than
the corresponding entities in 5GS, i.e., between gNB-CU-CP and the Session
Management Function (SMF), respectively, which further results in the
performance improvement in terms of control plane latency and resource
utilisation. Transposition of UE signalling handling functionality to
functions in service plane simplifies the protocols between the control pane
and the user plane such as Next Generation Application Protocol (NGAP) between
the CN control plane and RAN and F1AP between the RAN control plane (gNB-CU-
CP) and the RAN user plane (gNB-DU).

The existing 5GS uses the same type of signalling messages for all use cases.
However, it is possible to have different signalling requirements for
different use cases, e.g., the Internet of Things (IoT) and human users. The
proposed architecture may support this requirement by employing use case
specific signalling service functions. Our proposal can also support flexible
function deployment and chaining as various service functions, such as the PDU
session service function, mobility service function, registration service
function, and authentication service function, can be placed flexibly and
chained together to serve UEs.

An additional advantage towards signalling security is presented here. 3GPP
specification [18] highlights the exposed AMF which is vulnerable to replay
attacks of NAS signalling messages between the UE and AMF (control plane of
the CN). In a similar way, [19] presents the exposed RAN which is susceptible
to replay attacks to RRC signalling messages between the UE and RAN (gNB-CU-CP
(control plane of RAN)) as the Uu interface also carries sensitive RRC
signalling. Furthermore, the European Union Agency for Cybersecurity (ENISA)
[20], in its report recommends that the N2 interface between the 5GS RAN and
AMF is a target for attackers since they carry sensitive signalling between
the RAN and the CN. Therefore, in this context, the proposed architecture may
have some advantages towards the UE signalling security between the UE and the
signalling service function. Since UE signalling is segregated from the
control plane (of RAN and CN) and is terminated to a separate signalling
server, it leads to the possibility of localizing the attack originating from
a UE within the signalling server without compromising the network control
plane where the architectural and logical control and management of RAN and CN
are located. This segregation allows us to improve the UE-related signalling
security of future mobile networks.

##  IV Information Flow Comparison

In this section, we compare the information flows of the proposed architecture
and the existing 5GS architecture. We consider the PDU session establishment
and mobility services example to differentiate the working of the existing 5GS
and the proposed architectures.

Figure 2: Network entities, signalling and control message flow for PDU
session establishment in 5GS Figure 3: Network entities, signalling and
control message flow for PDU session establishment in the proposed
architecture

###  IV-A PDU session establishment as a service

Figure 4: PDU session establishment procedure in the proposed architecture

Fig. 2 and Fig. 3 show the entities involved in PDU session signalling for the
5GS and the proposed architecture, respectively. In 5GS, messages are
exchanged between UE and SMF for PDU session-related signalling via RAN (it
requires both gNB-DU and gNB-CU) and AMF. However, signalling messages are
directly exchanged between UE and the (PDU session service function (PSSF))
service function via RAN (it requires only RAN-DU) in the proposed
architecture, as shown in Fig. 3, which implies that in the existing 5GS,
signalling takes place through multiple hops. In contrast, the number of hops
is reduced in the proposed architecture. Further, the control plane collects
all requirements from the PSSF (which in turn are received by PSSF from the UE
as shown in Fig. 3) via the application-control interface and establishes the
PDU session.

The complete message sequences for establishing PDU sessions for the existing
5GS are detailed in [17] while simplified call flow for the proposed
architecture is shown in Fig. 4111In call flows and simulations, only those
messages are considered and compared which are different in proposed and
existing architectures. Please note that the controllers do not require
response messages from the resource (user) plane, as the controller knows
about user plane resource information; it handles resource decision-making.
Therefore, the proposed architecture eliminates many such messages. For
example, the N4 session modification request and response are exchanged
between SMF and UPF in 5GS architecture [17], while the session modification
command (message 3 in Fig. 4 and message 9 in Fig. 7) is exchanged between the
CN controller and CN user plane (UPF) in the proposed architecture. There is
no need for a session modification response message from the UPF. Hence, these
reductions in the messages simplify both the session establishment and
mobility procedure (to be discussed next). Please note that even though using
RAN-User Plane (RAN-UP) and other network functions/messages is necessary, we
have shown only the CN functions in the call flow to keep the analysis
tractable even though RAN functions will also be required in real systems.
However, keeping the RAN functions out of the call flows is not likely to
alter the conclusions drawn here. This note applies to mobility services also.

###  IV-B Mobility as a service

We consider mobility as another service to illustrate the difference between
the existing 5GS and the proposed architecture. Fig. 5 and Fig. 6 show the
network entities, signalling and control message flow of the existing 5GS and
proposed architecture, respectively. S-DU and T-DU represent source gNB-DU and
target gNB-DU, respectively. Similarly, the Source-Centralized Unit-User Plane
(S-CU-UP) and Target-Centralized Unit-User Plane (T-CU-UP) represent source
gNB-CU-UP and target gNB-CU-UP, respectively. S-CU-CP and T-CU-CP represent
source gNB-CU-CP and target gNB-CU-CP, respectively. Also, the interaction
between the RAN controller and the CN controller is through the inter-
controller interface, as shown in Fig. 6. Signalling takes place between UE
and MSF via S-DU before handover while after handover it is through T-DU.
Likewise, the data path between UE and UPF is by way of S-UP before handover
while it is via T-UP after handover.

Figure 5: Network entities, signalling and control message flow in case of
mobility service for the existing 5GS architecture Figure 6: Network entities,
signalling and control message flow in case of mobility service for the
proposed architecture

Mobility call flow for the existing 5GS is available in [17]. Fig. 7 shows the
mobility call flow which illustrates the handover procedure of the proposed
architecture. For the sake of simplicity, splitting S-UP into S-DU and S-CU-
UP, T-UP into T-DU and T-CU-UP is not shown. However, the reason behind the
simplification of mobility procedure/messages is the same as explained for PDU
session establishment in Section IV-B.

Figure 7: Mobility procedure in the proposed architecture

##  V System Model

TABLE I: system model for PDU session establishment PDU session establishment
---
PEPA Modules | Code Description
UE NF |  $Ue_{1}$ ${}_{=}^{def}$ ($acc_{uep}$, $r_{a}$).(process, $r_{iat}$).$Ue_{2}$
|  $Ue_{2}$ ${}_{=}^{def}$ ($req_{pduse}$, $r_{r}$).($rep_{pduse}$,
$r_{r}$).$Ue_{1}$
PSSF NF |  $Pssf_{1}$ ${}_{=}^{def}$ ($req_{pduse}$, $r_{r}$).$Pssf_{2}$
|  $Pssf_{2}$ ${}_{=}^{def}$ ($acc_{pssfp}$, $r_{a}$).(process,
$r_{p}$).$Pssf_{3}$
|  $Pssf_{3}$ ${}_{=}^{def}$ ($req_{sc}$, $r_{r}$).($rep_{sc}$,
$r_{r}$).$Pssf_{4}$
|  $Pssf_{4}$ ${}_{=}^{def}$ ($acc_{pssfp}$, $r_{a}$).(process,
$r_{p}$).$Pssf_{5}$
|  $Pssf_{5}$ ${}_{=}^{def}$ ($rep_{pduse}$, $r_{r}$).$Pssf_{1}$
CN Controller NF |  $Con_{1}$ ${}_{=}^{def}$ ($req_{sc}$, $r_{r}$).$Con_{2}$
|  $Con_{2}$ ${}_{=}^{def}$ ($acc_{conp}$, $r_{a}$).(process,
$r_{p}$).$Con_{3}$
|  $Con_{3}$ ${}_{=}^{def}$ ($req_{n4est}$, $r_{r}$).($rep_{n4est}$,
$r_{r}$).$Con_{4}$
|  $Con_{4}$ ${}_{=}^{def}$ ($acc_{conp}$, $r_{a}$).(process,
$r_{p}$).$Con_{5}$
|  $Con_{5}$ ${}_{=}^{def}$ ($rep_{sc}$, $r_{r}$).$Con_{1}$
UPF NF |  $Upf_{1}$ ${}_{=}^{def}$ ($req_{n4est}$, $r_{r}$).$Upf_{2}$
|  $Upf_{2}$ ${}_{=}^{def}$ ($acc_{upfp}$, $r_{a}$).(process,
$r_{p}$).$Upf_{1}$
UE Processor |  $Uep_{1}$ ${}_{=}^{def}$ ($acc_{uep}$, $r_{a}$).$Uep_{2}$
|  $Uep_{2}$ ${}_{=}^{def}$ (process, $r_{p}$).$Uep_{1}$
PSSF Processor |  $Pssfp_{2}$ ${}_{=}^{def}$ (process, $r_{p}$).$Pssfp_{1}$
CN Controller |  $Conp_{1}$ ${}_{=}^{def}$ ($acc_{conp}$, $r_{a}$).$Conp_{2}$
Processor |  $Conp_{2}$ ${}_{=}^{def}$ (process, $r_{p}$).$Conp_{1}$
UPF Processor |  $Upfp_{1}$ ${}_{=}^{def}$ ($acc_{upfp}$, $r_{a}$).$Upfp_{2}$
|  $Upfp_{2}$ ${}_{=}^{def}$ (process, $r_{p}$).$Upfp_{1}$
System Equation | ((($Ue_{1}$[n] ${}_{L_{1}}^{\bowtie}$ $Pssf_{1}$[$N_{pssf}$.$N_{pssfp}$.$N_{t}$])
|  ${}_{L_{2}}^{\bowtie}$ $Con_{1}$[$N_{con}$.$N_{conp}$.$N_{t}$])
|  ${}_{L_{3}}^{\bowtie}$ $Upf_{1}$[$N_{upf}$.$N_{upfp}$.$N_{t}$])
|  ${}_{L_{4}}^{\bowtie}$ ((($Uep_{1}$[n] ${}_{\phi}^{\bowtie}$
$Pssfp_{1}$[$N_{pssf}$.$N_{pssfp}$])
|  ${}_{\phi}^{\bowtie}$ $Conp_{1}$[$N_{con}$.$N_{conp}$])
|  ${}_{\phi}^{\bowtie}$ $Upfp_{1}$[$N_{upf}$.$N_{upfp}$])
Cooperation Set |  $L_{1}$ = $<$$req_{pduse}$, $rep_{pduse}$$>$
|  $L_{2}$ = $<$$req_{sc}$, $rep_{sc}$$>$
|  $L_{3}$ = $<$$req_{n4est}$$>$
|  $L_{4}$ = $<$$acc_{uep}$, $process$, $acc_{pssfp}$,
|  $acc_{conp}$, $acc_{upfp}$$>$
|  $\phi$ = $<>$
TABLE II: system model for mobility Mobility
---
PEPA Modules | Code Description
UE NF |  $Ue_{1}$ ${}_{=}^{def}$ ($acc_{uep}$, $r_{a}$).($measure$, $r_{iat}$).$Ue_{2}$
|  $Ue_{2}$ ${}_{=}^{def}$ ($reconfig$, $r_{r}$).$Ue_{3}$
|  $Ue_{3}$ ${}_{=}^{def}$ ($rachreq$, $r_{r}$).($rachres$, $r_{r}$).$Ue_{4}$
|  $Ue_{4}$ ${}_{=}^{def}$ ($reconfigcomp$,$r_{r}$).$Ue_{1}$
T-UP NF |  $Upt_{1}$ ${}_{=}^{def}$ ($pathsetup$, $r_{r}$).$Upt_{2}$
|  $Upt_{2}$ ${}_{=}^{def}$ ($acc_{uptp}$,
$r_{a}$).($process$,$r_{p}$).$Upt_{3}$
|  $Upt_{3}$ ${}_{=}^{def}$ ($rachreq$,$r_{r}$).($rachres$,$r_{r}$).$Upt_{1}$
MSF NF |  $Msf_{1}$ ${}_{=}^{def}$ ($measure$,$r_{r}$).$Msf_{2}$
|  $Msf_{2}$ ${}_{=}^{def}$ ($acc_{msfp}$,$r_{a}$).($horeq$,$r_{r}$).$Msf_{3}$
|  $Msf_{3}$ ${}_{=}^{def}$ ($hores$,$r_{r}$).$Msf_{4}$
|  $Msf_{4}$ ${}_{=}^{def}$
($acc_{msfp}$,$r_{a}$).($reconfig$,$r_{r}$).$Msf_{5}$
|  $Msf_{5}$ ${}_{=}^{def}$ ($reconfigcomp$,$r_{r}$).$Msf_{6}$
|  $Msf_{6}$ ${}_{=}^{def}$ ($acc_{msfp}$,$r_{a}$).
| ($pathswitch$,$r_{r}$).$Msf_{1}$
RAN Controller NF |  $Ran_{1}$ ${}_{=}^{def}$ ($horeq$,$r_{r}$).$Ran_{2}$
|  $Ran_{2}$ ${}_{=}^{def}$ ($acc_{ranp}$,$r_{a}$).($pathsetup$,$r_{r}$)
| .($hores$,$r_{r}$).$Ran_{1}$
CN Controller NF |  $Cn_{1}$ ${}_{=}^{def}$ ($pathswitch$,$r_{r}$).$Cn_{2}$
|  $Cn_{2}$ ${}_{=}^{def}$ ($acc_{cnp}$,$r_{a}$).($session$,$r_{r}$).$Cn_{1}$
UPF NF |  $Upf_{1}$ ${}_{=}^{def}$ ($session$,$r_{r}$).$Upf_{2}$
|  $Upf_{2}$ ${}_{=}^{def}$
($acc_{upfp}$,$r_{a}$).($process$,$r_{p}$).$Upf_{1}$
UE Processor |  $Uep_{1}$ ${}_{=}^{def}$ ($acc_{uep}$,$r_{a}$).$Uep_{2}$
|  $Uep_{2}$ ${}_{=}^{def}$ ($measure$,$r_{iat}$).$Uep_{1}$
T-UP Processor |  $Uptp_{1}$ ${}_{=}^{def}$ ($acc_{uptp}$,$r_{a}$).$Uptp_{2}$
|  $Uptp_{2}$ ${}_{=}^{def}$ ($rachreq$,$r_{r}$).$Uptp1$
| +($rachres$,$r_{r}$).$Uptp_{1}$
MSF Processor |  $Msfp_{1}$ ${}_{=}^{def}$ ($acc_{msfp}$,$r_{a}$).$Msfp_{2}$
|  $Msfp_{2}$ ${}_{=}^{def}$ ($horeq$,$r_{r}$).$Msfp_{1}$+($reconfig$,$r_{r}$)
| .$Msfp_{1}$+($pathswitch$,$r_{r}$).$Msfp_{1}$
RAN Processor |  $Ranp_{1}$ ${}_{=}^{def}$ ($acc_{ranp}$,$r_{a}$).$Ranp_{2}$
|  $Ranp_{2}$ ${}_{=}^{def}$
($pathsetup$,$r_{r}$).($hores$,$r_{r}$).$Ranp_{1}$
CN Processor |  $Cnp_{1}$ ${}_{=}^{def}$ ($acc_{cnp}$,$r_{a}$).$Cnp_{2}$
|  $Cnp_{2}$ ${}_{=}^{def}$ ($session$,$r_{r}$).$Cnp_{1}$
UPF Processor |  $Upfp_{1}$ ${}_{=}^{def}$ ($acc_{upfp}$,$r_{a}$).$Upfp_{2}$
|  $Upfp_{2}$ ${}_{=}^{def}$ ($session$,$r_{r}$).$Upfp_{1}$
System Equation | ((((($Ue_{1}$[n]${}_{L_{1}}^{\bowtie}$$Upt_{1}$[$N_{upt}$.$N_{uptp}$.$N_{t}$])
|  ${}_{L_{2}}^{\bowtie}$$Msf1$[$N_{msf}$.$N_{msfp}$.$N_{t}$])
|  ${}_{L_{3}}^{\bowtie}$$Ran_{1}$[$N_{ran}$.$N_{ranp}$.$N_{t}$])
|  ${}_{L_{4}}^{\bowtie}$$Cn_{1}$[$N_{cn}$.$N_{cnp}$.$N_{t}$])
|  ${}_{L_{5}}^{\bowtie}$$Upf_{1}$[$N_{upf}$.$N_{upfp}$.$N_{t}$])
|
${}_{L_{6}}^{\bowtie}$((((($Uep_{1}$[n]${}_{\phi}^{\bowtie}$$Uptp_{1}$[$N_{upt}$.$N_{uptp}$])
|  ${}_{\phi}^{\bowtie}$$Msfp_{1}$[$N_{msf}$.$N_{msfp}$])
|  ${}_{\phi}^{\bowtie}$$Ranp_{1}$[$N_{ran}$.$N_{ranp}$])
|  ${}_{\phi}^{\bowtie}$$Cnp_{1}$[$N_{cn}$.$N_{cnp}$])
|  ${}_{\phi}^{\bowtie}$$Upfp_{1}$[$N_{upf}$.$N_{upfp}$])
Cooperation Set |  $L_{1}$ = $<rachreq,rachres>$
|  $L_{2}$ = $<$$measure$, $reconfig$,
|  $reconfigcomp$$>$
|  $L_{3}$ = $<pathsetup,horeq,hores>$
|  $L_{4}$ = $<pathswitch>$
|  $L_{5}$ = $<session>$
|  $L_{6}$ = $<$$acc_{uep}$, $acc_{uptp}$, $acc_{msfp}$,
|  $acc_{ranp}$, $acc_{cnp}$, $acc_{upfp}$$>$
|  $\phi$ = $<>$

This section presents the system model for the proposed architecture using
PEPA. PEPA is a formal high-level language for the quantitative modelling of a
distributed system [13]. Table I and Table II represent the system model for
the proposed architecture for the PDU session establishment and mobility
procedures, respectively. To explain the system model, we use the PDU session
establishment (or session establishment) procedure (shown in Fig. 4).

The session establishment procedure requires PSSF, CN controller and UPF as
the key CN functions in the proposed architecture. These NFs are modelled as
PEPA components. In addition, a UE is also modelled as a PEPA component. Each
PEPA component (representing UE or a CN NF) goes through a set of states
during the handling of the procedure. The individual component states are
denoted by associating a unique number with the name of the component (e.g.,
$Pssf_{1}$, represents the first state of component, PSSF). Each component
performs a set of actions, such as accessing the processor or sending a
request/response. These actions are denoted in lowercase, and subscripts are
added to provide further distinction (as $action_{actiondetail}$). For
example, the notation for the action of PDU session establishment request and
response can be $req_{pduse}$ and $rep_{pduse}$, respectively. Each action is
associated with a specific rate value, $r$. The rate (number of actions
performed per unit time) models the expected duration of the action in the
PEPA component and is taken as reference from [21], [22] and [23].

Let us now understand the details of modelling of NF states as shown in Table
I. Consider UE as an example. The UE acquires the processor in its initial
state ($acc_{uep}$, $r_{a}$) and performs the processing action ($process$,
$r_{iat}$) before sending a request. The second state, $Ue_{2}$, models the
request ($req_{pduse}$, $r_{r}$) and response ($rep_{pduse}$, $r_{r}$)
messages exchanged between UE and PSSF for the PDU session establishment. NFs
acquire processors to process a request/response. In Table I, UEP, PSSFP, CONP
and UPFP are the processing entities for UE, PSSF, CN controller (CON) and UPF
respectively. These processing entities are modelled such that each NF
processor has two states. For instance, the first state of UEP, $Uep_{1}$, is
for acquiring the processor ($acc_{uep}$), and the second state, $Uep_{2}$,
performs the processing action ($process$). Similarly, the other NFs and their
processing entities are modelled.

As discussed in this section, the system model uses the following additional
parameters: $n$ denotes the number of UEs; $N_{pssf}$, $N_{con}$, and
$N_{upf}$ are the number of NF instances for PSSF, CN controller (CON), and
UPF, respectively. Similarly, $N_{pssfp}$, $N_{conp}$, and $N_{upfp}$ are the
number of PSSF processor (PSSFP), CN controller processor (CONP) and UPF
processor (UPFP), respectively. Please note that each processor can handle a
set of concurrent threads, $N_{t}$. Thus, the product
$N_{nf}$·$N_{nfp}$·$N_{t}$ (as mentioned in the system model equation)
represents the total number of threads for a type of NF. Moreover, the product
$N_{nf}$·$N_{nfp}$ is the total number of processors allocated to a type of
NF, e.g., for UPF processor.

The system equation represents the overall system model. The cooperation
operator (“${\bowtie}$”), for example, A ${}_{L}^{\bowtie}$ B, models the
interactions between NFs A and B over the actions defined in the cooperation
set $L$. It can be noted that it is possible that component A
${}_{L}^{\bowtie}$ B will have different behaviour from component A
${}_{K}^{\bowtie}$ B if L$\neq$K. Let us consider an example from Fig. 4,
where PSSF and CN controller (CON) interact with each other for session
context request/response $req_{sc}$/$rep_{sc}$. These actions are defined in
cooperation set $L_{2}$, as shown in Table I. Therefore, the system equation
$Pssf_{1}$[$N_{pssf}$.$N_{pssfp}$.$N_{t}$] ${}_{L_{2}}^{\bowtie}$
$Con_{1}$[$N_{con}$.$N_{conp}$.$N_{t}$], models the interaction between PSSF
and CN controller over the cooperation set $L_{2}$. In a similar way, the
overall system equation, as shown in Table I and Table II represents the
interaction between the various NFs as shown in the two call flows, Fig. 4 and
Fig. 7, respectively.

##  VI performance evaluation

This section presents the performance comparison between the existing 5GS and
the proposed architecture analysed using the PEPA Eclipse plug-in [24], a
software tool integrated into the popular Eclipse platform. This tool supports
various performance measures [22] as discussed below, which help evaluate the
network’s performance.

  1. 1.

Session establishment rate (or the number of successful handovers in the case
of mobility): The number of session establishments are measured for the action
(say, $rep_{pduse}$, which describes the completion of the session
establishment procedure), representing the session establishment rate.
Similarly, the number of successful handovers is measured for the action
‘$session$’(as performed by UPF NF in Table II), which signifies the
completion of the handover procedure.

  2. 2.

Average response time: It measures the UE waiting time for any specific
request and reflects the system’s operating speed. We consider the average
response time as the duration of the completion of the session establishment
procedure. Similarly, we consider the mobility procedure’s average response
time as the completion of the handover procedure.

  3. 3.

Utilisation: Utilisation measures the NFs processor capacity utilised during
the procedure. The utilisation of any NF (for example, PSSF processor) is
derived from its population level (one of the features available in the tool)
while performing any process.

  4. 4.

Scalability: Scalability (S), in simple terms, measures a network’s ability to
increase or decrease its size, performance and cost in response to changes in
system processing demands. Alternatively, according to Equation 1, scalability
can be defined as the ratio between the productivity of a system at two
configurations (configuration here implies the number of NFs used) having
different scales, say $m_{1}$ and $m_{2}$ [25], which corresponds to the
different numbers of NFs used in the network, say $m_{1}$ = (1,1,1) and
$m_{2}$ = (3,3,1). The mathematical expression for scalability is given as
[25]:

$S(m_{1},m_{2})=\frac{C(m_{2})}{C(m_{1})}$ (1)

Where, C(m) is the productivity of a system at the scale m, given by (Equation
2):

$C(m)=\frac{t(m)\cdot r(m)}{U(m)}$ (2)

Where t(m) is the average number of sessions established at scale m, U(m) is
the processor utilisation of the system at scale m, and r(m) (Equation 3) is
determined by evaluating the response time performance of the scaled system.
We consider the following equation [25] to evaluate the performance function
r(m) by using the average response time T(m), at scale m, with the target
average response time T [22].

$r(m)=\frac{1}{1+T(m)/T}$ (3)

###  VI-A Results and Analysis

In this section, we present the performance results for 5GS and the proposed
architecture in the case of PDU session establishment service and mobility
service.

####  VI-A1 PDU Session Establishment Service

The performance analysis of the proposed architecture and the existing 5GS for
the session establishment procedure is discussed in this section. Fig. 8 and
Fig. 9 show the session establishment rate with respect to the number of UEs
for 5GS and the proposed architecture using two different configurations. For
instance, ($N_{pssf}$, $N_{con}$, $N_{upf}$) = (1,1,1) for the proposed
architecture is the basic configuration with single NF assigned and
($N_{pssf}$, $N_{con}$, $N_{upf}$) = (3,3,1) is the configuration for a scaled
system with three NFs assigned to PSSF and CON while one to UPF. Similarly,
basic and the scaled configuration for 5GS is defined as ($N_{amf}$,
$N_{smf}$, $N_{upf}$) = (1,1,1) and ($N_{amf}$, $N_{smf}$, $N_{upf}$) =
(3,3,1), respectively.

Figure 8: Session establishment (number of sessions per unit time) for the
proposed and the 5GS architecture having the basic configuration Figure 9:
Session establishment (number of sessions per unit time) for the proposed and
the 5GS architecture having the scaled configuration Figure 10: Processor
utilisation of session establishment for the proposed and the 5GS architecture
having the basic configuration Figure 11: Processor utilisation of session
establishment for the proposed and the 5GS architecture having scaled
configuration Figure 12: Scalability of PDU session for 5GS and the proposed
architecture

Results show that the proposed architecture can achieve a higher session
establishment rate compared to the existing 5GS in case of both basic and
scaled configurations. Although the session establishment rate has increased
using a scaled configuration for proposed and existing architectures compared
to the session establishment rate achieved using a basic configuration, the
proposed architecture has achieved a higher session establishment rate than
the 5GS. The saturation point for existing 5GS, as shown in Fig. 8, is around
10,000 users i.e. it can serve a maximum number of 10,000 users, while the
session establishment rate for the proposed architecture saturates at around
20,000 users. Similarly, Fig. 9 shows that 5GS saturates at around 34,000
users. As the saturation point is reached, the network drops the incoming
requests from the users. This means that with the given number of
processors/NFs, the proposed architecture can achieve a higher session
establishment rate. In contrast, more processors/NFs are required to support
more number of session establishments.

The processor utilisation for all the NFs of the existing 5GS and the proposed
architecture for basic and the scaled configuration is shown in Fig. 10 and
Fig. 11, respectively. For instance, the PSSFP reaches its maximum utilisation
explaining the saturation point for the session establishment rate. Although
at this point, CONP and UPFP are not fully utilised. These results show that
the request processing chain fails if an NF becomes a bottleneck for the
consecutive chain.

Scalability for the existing 5GS and the proposed architecture is evaluated
based on Equation 1. It is plotted in Fig. 12 based on the results obtained
for session establishment rate, average response time and utilisation from the
PEPA-based simulation and modelling. As stated earlier, we consider the
following two configurations $m_{1}$ and $m_{2}$ for estimating the
scalability metric. Fig. 12 shows that the existing 5GS can serve 10,000 users
for a basic configuration, and the proposed architecture can serve 20,000
users. Similarly, the existing 5GS reaches its saturation point at 34,000
users, and the proposed architecture saturates at 62,000 users for scaled
configuration. Therefore, it implies that the proposed architecture performs
better and can serve more users than the existing 5GS. Besides, the proposed
is more scalable with increased users for the same number of NFs/processors.
Please note that a similar explanation for all the performance measures
(successful handovers, processor utilization and scalability) holds in the
case of mobility service.

####  VI-A2 Mobility Service

Figure 13: Number of successful handovers per unit time for the proposed and
the 5GS architecture having the basic configuration Figure 14: Number of
successful handovers per unit time for the proposed and the 5GS architecture
having the scaled configuration Figure 15: Processor Utilisation in case of
mobility for the proposed and the 5GS architecture having the basic
configuration Figure 16: Processor Utilisation in case of mobility for the
proposed and the 5GS architecture having the scaled configuration Figure 17:
Scalability in case of Mobility

This section presents the comparative analysis of the existing 5GS and the
proposed architecture for the mobility service. Similar to the session
establishment, the analysis is performed for the basic and the scaled
configurations. Therefore, the basic configuration for the proposed
architecture is given as ($N_{upt}$, $N_{msf}$, $N_{ran}$, $N_{cn}$,
$N_{upf}$) = (1,2,2,1,1) and for the 5GS architecture is ($N_{sdu}$,
$N_{scu}$, $N_{tdu}$, $N_{tcu}$, $N_{amf}$, $N_{smf}$, $N_{upf}$)  =
(1,1,1,1,1,1,1). Similarly, the scaled configuration for the proposed
architecture is ($N_{upt}$, $N_{msf}$, $N_{ran}$, $N_{cn}$, $N_{upf}$) =
(3,6,6,3,3) and for the 5GS architecture is given as ($N_{sdu}$, $N_{scu}$,
$N_{tdu}$, $N_{tcu}$, $N_{amf}$, $N_{smf}$, $N_{upf}$)  = (3,3,3,3,3,3,3).
Here $N_{upt}$, $N_{msf}$, $N_{ran}$, $N_{cn}$, $N_{upf}$ are the number of
Target-User Plane (T-UP), MSF, RAN controller, CN controller and UPF
respectively in the system model. Similarly, $N_{sdu}$, $N_{scu}$, $N_{tdu}$,
$N_{tcu}$, $N_{amf}$, $N_{smf}$, $N_{upf}$ are the number of S-DU, S-CU, T-DU,
T-CU, AMF, SMF, and UPF respectively. Please note that for brevity, we have
not split S-CU into S-CU-CP and S-CU-UP and T-CU into T-CU-CP and T-CU-UP
while modelling the mobility call flow procedure for the 5GS. Further, we
provide an equal number of functions and associated processors to the 5GS and
the proposed architecture for justified comparison.

After reaching the saturation point, the system starts to drop handovers. Fig.
13 and Fig. 14 show that the proposed architecture serves more successful
handovers per unit time compared to the existing 5GS for both the basic and
the scaled configurations, respectively. The saturation point for the existing
5GS is 20,000 users, while for the proposed, the saturation is 30,000 users
for the basic configuration. Similarly, the saturation point for the existing
5GS is around 60,000 users, while for the proposed, the saturation is around
90,000 users for the scaled configuration. The number of successful handovers
per unit of time has increased using a scaled configuration for both
architectures.

Fig. 15 and Fig. 16 are the result of processor utilisation for both the 5GS
and the proposed architecture. Fig. 17 shows the scalability results in the
case of mobility service for 5GS and the proposed architectures. It can be
observed from the scalability results that 5GS reaches its saturation point
earlier than the proposed architecture and the proposed architecture is more
scalable.

##  VII CONCLUSION AND FUTURE WORK

In this paper, we have proposed a novel mobile network architecture for
separating the UE signalling from the network control functionality, enhancing
the modularity, scalability, and flexibility of the network. The transposition
of UE signalling functionality to service functions leads to simplified
protocols and opens up ways to implement use case specific signalling in
mobile networks. The proposed architecture also has improved alignment with
the SDN principles.

We have considered PDU session establishment and mobility services as examples
to analyse the performance of the proposed architecture using the PEPA-based
simulation method. Based on the performance results and other benefits, it can
be concluded that the proposed architecture is a promising option for future
networks to handle vast and diverse traffic demands. We plan to extend this
work to analyse other features/services of mobile networks, such as
authentication, network slicing, development of protocols between (signalling)
service functions and the control plane, and addressing security threats in
the 6GS mobile network (touched upon in section III) in future.

## Acknowledgment

We acknowledge the Ministry of Electronics and Information Technology (MeitY),
India, for supporting the project.

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# A geometric take on Kostant’s Convexity Theorem

Ricardo A. E. Mendes  University of Oklahoma
Department of Mathematics
601 Elm Ave
Norman, OK, 73019-3103, USA<EMAIL_ADDRESS>
###### Abstract.

Given a compact Lie group $G$ and an orthogonal $G$-representation $V$, we
give a purely metric criterion for a closed subset of the orbit space $V/G$ to
have convex pre-image in $V$. In fact, this also holds with the natural
quotient map $V\to V/G$ replaced with an arbitrary submetry $V\to X$.

In this context, we introduce a notion of “fat section” which generalizes
polar representations, representations of non-trivial copolarity, and
isoparametric foliations.

We show that Kostant’s Convexity Theorem partially generalizes from polar
representations to submetries with a fat section, and give examples
illustrating that it does not fully generalize to this situation.

###### Key words and phrases:

Orbit space, submetry, convexity, polar action

###### 2020 Mathematics Subject Classification:

57S15, 51F99

The author has been supported by the NSF grant DMS-2005373

##  1\. Introduction

B. Kostant’s celebrated “Convexity Theorem” [Kos73, Theorem 8.2] can be
phrased as follows:

######  Theorem 1 (Kostant).

Let $V$ be a real orthogonal representation of the compact group $G$, with
connected orbits. Assume the representation is polar, with section
$\Sigma\subset V$ and Weyl group $W$ acting on $\Sigma$. Then

$\pi_{\Sigma}(G\cdot v)=\operatorname{conv}(W\cdot v)$

for all $v\in\Sigma$. Here $\pi_{\Sigma}$ denotes the orthogonal projection
onto $\Sigma$, $\operatorname{conv}(\cdot)$ the convex hull, $G\cdot v$ the
$G$-orbit through $v$, and $W\cdot v=(G\cdot v)\cap\Sigma$ the $W$-orbit
through $v$.

Recall that the $G$-representation $V$ is called polar with section $\Sigma$
if $\Sigma$ is a linear subspace of $V$ that intersects all $G$-orbits
orthogonally (see [Dad85]). The formulation above is equivalent to the
original because, up to having the same orbits, the class of polar
representations with connected orbits coincides with the class of isotropy
representations of symmetric spaces (see [Dad85, Proposition 6]). Two notable
special cases are the adjoint representation of a connected compact Lie group
$G$ on its Lie algebra $V$, with $\Sigma$ the Lie algebra of a maximal torus
and $W$ the usual Weyl group; and the Schur–Horn Theorem concerning the
diagonal entries of symmetric matrices with a given set of eigenvalues (see
[LRT99, page 150]).

An important consequence is that the study of $G$-invariant convex subsets of
$V$ is in some sense reduced to the study of $W$-invariant convex subsets of
$\Sigma$:

######  Corollary 2.

With assumptions and notations as in Theorem 1:

  1. (a)

For every $G$-invariant convex subset $K\subset V$,

$\pi_{\Sigma}(K)=K\cap\Sigma.$

  2. (b)

The map $K\mapsto\pi_{\Sigma}(K)=K\cap\Sigma$ is a bijection between
$G$-invariant convex subsets of $V$ and $W$-invariant convex subsets of
$\Sigma$.

For a proof of (a), see e.g. [KS20, Corollary 3.4]. The proof of (b) is
straightforward (see proof of Theorem C below for a generalization).

In terms of quotient spaces, subsets of $V/G$ (resp. $\Sigma/W$) correspond to
$G$-invariant subsets of $V$ (resp. $W$-invariant subsets of $\Sigma$). Thus
Corollary 2(b) states that the isometry $\Sigma/W\to V/G$ induced by the
inclusion $\Sigma\to V$ preserves the collection of subsets that have convex
pre-images in $\Sigma,V$. This is an immediate consequence of our first main
result, which gives a _purely metric_ criterion for a closed subset $S\subset
V/G$ to have convex pre-image in $V$, for _any_ (not necessarily polar)
$G$-representation $V$. More generally, one can also replace the map $V\to
V/G$ with any submetry, that is, any map $V\to X$ to a metric space $X$ that
takes metric balls to metric balls of the same radius (see Subsection 2.1
below):

######  Theorem A.

Let $V$ be a finite-dimensional real vector space with an inner product, $X$ a
metric space, and $\sigma\colon V\to X$ a submetry. Let $S\subset X$ be a
closed subset, and denote by $f\colon X\setminus S\to(0,\infty)$ the distance
function to $S$.

Then $\sigma^{-1}(S)$ is a convex subset of $V$ if and only if

$\lim\sup_{y\to x}\frac{f(y)-f(x)}{d(y,x)}=1$

for every $x\in X\setminus S$.

The condition given above on the function $f$ means that its gradient has norm
one at every point of $X\setminus S$. This can be made precise in Alexandrov
Geometry, see Section 3.

Theorem A fits in a general theme: the interplay between the geometry of the
isometric $G$-action on $V$ (or, more generally, some Riemannian manifold) and
the metric geometry of the orbit space $V/G$. This is a traditional topic
explored extensively in the literature, for example in [HK89, Gro02, GL14,
GLLM19, Men20, GKW21].

Going beyond the polar case, whenever a $G$-representation $V$ admits a
“reduction”, that is, a $G^{\prime}$-representation $V^{\prime}$ such that
$V/G$ is isometric to $V^{\prime}/G^{\prime}$ (called “quotient-equivalent” in
[GL14]), Theorem A implies that there is a bijection between the respective
closed invariant convex subsets. More generally:

######  Corollary B.

Let $V,V^{\prime}$ be finite-dimensional real vector spaces with inner
product, $X,X^{\prime}$ metric spaces, and $\sigma\colon V\to X$ and
$\sigma^{\prime}\colon V^{\prime}\to X^{\prime}$ be submetries. Let
$\varphi\colon X\to X^{\prime}$ be an isometry, and $S\subset X$ be a closed
subset. Then $\sigma^{-1}(S)$ is convex if and only if
$(\sigma^{\prime})^{-1}(\varphi(S))$ is convex. Thus $\varphi$ induces a
bijection between closed convex $\sigma$-saturated subsets of $V$ and closed
convex $\sigma^{\prime}$-saturated subsets of $V^{\prime}$.

Having generalized Corollary 2(b), we investigate the extent to which Theorem
A can be used to generalize Theorem 1 and Corollary 2(a). Since these concern
the orthogonal projection $\pi_{\Sigma}$, we need the “reduction” to be
induced by a vector subspace $\Sigma\subset V$. More precisely, we introduce
the following definition: Given a submetry $\sigma\colon V\to X$, such that
$\\{0\\}$ is a fiber, we call a subspace $\Sigma\subset V$ a _fat section_
111term borrowed from [Mag09, Mag10]. for $\sigma$ if
$\sigma|_{\Sigma}\colon\Sigma\to X$ is a submetry. Besides polar
representations, this generalizes a number of other objects, including
representations of non-trivial copolarity, principal isotropy group
reductions, and isoparametric foliations (see Section 4). Our second main
result is:

######  Theorem C.

Let $\sigma\colon V\to X$ be a submetry such that $\\{0\\}$ is a fiber, with
fat section $\Sigma\subset V$. Denote by $\pi_{\Sigma}$ the orthogonal
projection onto $\Sigma$. Then:

  1. (a)

For any $\sigma$-fiber $F\subset V$,

$\operatorname{conv}(F)\cap\Sigma=\pi_{\Sigma}(\operatorname{conv}(F))=\operatorname{conv}(F\cap\Sigma).$

In particular,

$\pi_{\Sigma}(F)\subset\operatorname{conv}(F\cap\Sigma).$

  2. (b)

For every $\sigma$-saturated convex set $K$,

$\pi_{\Sigma}(K)=K\cap\Sigma.$

  3. (c)

The map $K\mapsto\pi_{\Sigma}(K)=K\cap\Sigma$ is a bijection between
$\sigma$-saturated convex subsets of $V$ and $\sigma|_{\Sigma}$-saturated
convex subsets of $\Sigma$.

Thus Corollary 2 holds in this more general situation, as does one of the
inclusions in the statement of Theorem 1. As for the reverse inclusion, that
is, the equality “$\pi_{\Sigma}(F)=\operatorname{conv}(F\cap\Sigma)$” (or,
equivalently, convexity of $\pi_{\Sigma}(F)$), it does hold for isoparametric
foliations, see [Ter86]. Beyond this class, it is easy to find counter-
examples, see Section 5.

We finish with a couple of open questions:

######  Question 3.

Is there a submetry with fat section, that is not isoparametric, and for which
$\pi_{\Sigma}(F)=\operatorname{conv}(F\cap\Sigma)$ for every $\sigma$-fiber
$F\subset V$?

######  Question 4.

In the situation of Corollary B, does the bijection between $\sigma$-saturated
closed convex subsets of $V$ and $\sigma^{\prime}$-saturated closed convex
subsets of $V^{\prime}$ preserve some special class of convex sets, such as
spectrahedra or spectrahedral shadows?

Special cases of Question 4 have received much attention recently (see, for
example, [KS20, Kum21, SS20]).

### Acknowledgements

It is a pleasure to thank Alexander Lytchak for help with Alexandrov Geometry,
and for suggesting an earlier version of Theorem A. I am also grateful to
Marco Radeschi for suggesting a proof (very similar to the one presented here)
of Proposition 13(b), that convex hulls of saturated sets are saturated.

##  2\. Preliminaries

###  2.1. Submetries

The definition of _submetry_ goes back to [Ber87] (see also [KL20]): A
_submetry_ is a map $\sigma\colon Y\to X$ between metric spaces that maps
closed balls to closed balls of the same radius. The fibers of $\sigma$
(sometimes also called _leaves_) form a decomposition of $Y$ into pairwise
_equidistant_ closed subsets, in the sense that
$d(x,F^{\prime})=d(F,F^{\prime})$ for every two fibers $F,F^{\prime}$ and all
$x\in F$.

Conversely, given such a partition $\mathcal{F}$ of the metric space $Y$ (into
“leaves”), there is a unique metric on the set of leaves $X$ such that the
natural map $Y\to X$ is a submetry. Endowed with this metric, $X$ is called
the _leaf space_.

A function on $Y$ is called _$\sigma$ -basic_ (or just basic, if $\sigma$ is
clear from context) if it is constant on the fibers of $\sigma$, that is, if
it descends to a function on the “base” $X$. A subset of $Y$ is called
_$\sigma$ -saturated_ (or just saturated) if it is the union of
$\sigma$-fibers, that is, if it is the inverse image under $\sigma$ of a
subset of $X$.

The main source of examples of submetries are isometric group actions. Namely,
if the group $G$ acts on the metric space $Y$ by isometries and with closed
orbits, then the natural quotient map $Y\to Y/G$ is a submetry. The fibers
(“leaves”) are the $G$-orbits, the saturated subsets of $Y$ are the
$G$-invariant subsets, and the basic functions on $Y$ are the $G$-invariant
functions.

In the present paper, we consider submetries $V\to X$, where $V$ will always
denote a finite-dimensional real vector space with inner product (and
associated Euclidean metric). We mention a few structure results that apply to
this situation: $X$ is an Alexandrov space with non-negative curvature (see
[BGP92, 4.6]); every fiber is a subset of positive reach; and most fibers are
$C^{1,1}$-submanifolds (see [KL20] for the last two, and much more). If one
adds the assumption that every fiber is a smooth submanifold, one arrives at
the notion of “manifold submetry”, see [MR20c, MR20b] for structure results.

We will many times add the assumption that the singleton $\\{0\\}$ is a
$\sigma$-fiber, where $0\in V$ denotes the origin. It implies the following
version of the Homothetic Transformation Lemma (see also [Mol88, Lemma 6.2],
and [MR20c, Lemma 24]). The importance of such submetries is that they model
the “infinitesimal behavior” of more general submetries (compare [KL20,
Sections 5,7]).

######  Lemma 5.

Let $\sigma\colon V\to X$ be a submetry such that $\\{0\\}$ is a fiber. If
$v,w\in V$ are such that $\sigma(v)=\sigma(w)$, then $\sigma(\lambda
v)=\sigma(\lambda w)$ for all $\lambda\geq 0$.

###### Proof.

The ray $t\mapsto tv$ (respectively $t\mapsto tw$), for $t\geq 0$, minimizes
distance between $\\{0\\}$ and each fiber of the form
$\sigma^{-1}(\sigma(t_{0}v))$ (respectively $\sigma^{-1}(\sigma(t_{0}w))$),
for $t_{0}\geq 0$. Thus they descend to geodesic rays $\gamma_{v},\gamma_{w}$
in $X$. Recall that, $X$ being an Alexandrov space, geodesics do not branch,
and, since $\gamma_{v}(0)=\gamma_{w}(0)$ and $\gamma_{v}(1)=\gamma_{w}(1)$, we
obtain $\gamma_{v}=\gamma_{w}$ (see [BBI01, Exercises 10.1.2, 10.1.5]). In
particular, $\gamma_{v}(\lambda)=\gamma_{w}(\lambda)$, or, in other words,
$\sigma(\lambda v)=\sigma(\lambda w)$. ∎

Given a submetry $\sigma\colon V\to X$, if the singleton $\\{v\\}$ is a fiber,
we will say that $v$ is a _fixed point_ (of $\sigma$).

We will need the following Lemma, which follows from [KL20, Proposition 5.6],
and is also a slight generalization of a well-known fact about singular
Riemannian foliations (see [MR20a, Proposition 5]). For completeness we
provide an elementary proof.

######  Lemma 6.

Let $\sigma\colon V\to X$ be a submetry such that the origin $0$ is a fixed
point. Then the set $V_{0}$ of all fixed points is a vector subspace of $V$,
and $\sigma$ “splits” in the sense that $X$ is isometric to $V_{0}\times
X^{\prime}$ for some metric space $X^{\prime}$, and

$\sigma=\operatorname{Id}_{V_{0}}\times\sigma^{\prime}\colon V=V_{0}\times
V_{0}^{\perp}\to V_{0}\times X^{\prime}=X.$

Here $\sigma^{\prime}\colon V_{0}^{\perp}\to X^{\prime}$ is a submetry whose
unique fixed point is the origin.

We first give a separate statement of a basic fact from Euclidean Geometry
that will be useful in the proof of Lemma 6 and elsewhere:

######  Lemma 7.

Let $V$ be a vector space with inner product. Then, for $u,v\in V$ with
$\|u\|=1$, one has:

$\langle
v,u\rangle=\sup_{t>0}\left(t-d\left(v,tu\right)\right)=\lim_{t\to\infty}\left(t-d\left(v,tu\right)\right)$

###### Proof of Lemma 6.

We use induction on $\dim(V)$. If $\dim(V)=0$, there is nothing to prove.
Assume $\dim(V)>0$.

If the origin is the only fixed point, there is nothing to prove.

If $v\neq 0$ is a fixed point, consider the linear function $\lambda_{v}\colon
V\to\mathds{R}$ given by $\lambda_{v}(x)=\langle x,v\rangle$. We claim that
$\lambda_{v}$ is basic.

Indeed, assume $x,x^{\prime}\in V\setminus\\{0\\}$ belong to the same fiber.
Then $\|x\|=\|x^{\prime}\|$ by equidistance of fibers (because the origin is a
fiber), and so $\sigma(tx/\|x\|)=\sigma(tx^{\prime}/\|x^{\prime}\|)$ for all
$t>0$ by the Homothetic Transformation Lemma (Lemma 5). Again by equidistance
of fibers, we obtain
$d\left(v,tx/\|x\|\right)=d\left(v,tx^{\prime}/\|x^{\prime}\|\right)$ for all
$t>0$ (because $\\{v\\}$ is a fiber, by assumption). But Lemma 7 yields

$\lambda_{v}(x)=\|x\|\sup_{t>0}\left(t-d\left(v,tx/\|x\|\right)\right)$

and similarly for $x^{\prime}$. Therefore
$\lambda_{v}(x)=\lambda_{v}(x^{\prime})$.

Since $\lambda_{v}$ is basic, the level sets are saturated. Given a fiber
$F\subset\lambda_{v}^{-1}(0)$, it follows from equidistance of fibers that the
translate $cv+F$ is again a fiber, for every $c\in\mathds{R}$ (see [MR20a,
Prop. 5, proof of (b)$\implies$(a)] for more details). Therefore $\sigma$
splits as
$\operatorname{Id}_{\mathds{R}}\times\sigma^{\prime}\colon\mathds{R}\times\lambda_{v}^{-1}(0)\to
X=\mathds{R}\times X^{\prime}$ for some submetry
$\sigma^{\prime}\colon\lambda_{v}^{-1}(0)\to X^{\prime}$. Applying the
inductive hypothesis to $\sigma^{\prime}$ finishes the proof. ∎

Regarding convex sets, we will use the elementary observation:

######  Lemma 8.

Let $\sigma\colon V\to X$ be a submetry such that the origin $0$ is a fixed
point. Then every closed convex $\sigma$-saturated set $K$ has a fixed point.

###### Proof.

Take the point $v\in K$ closest to the origin (existence and uniqueness follow
from the assumption that $K$ is closed and convex). Then $\\{v\\}$ is
saturated, hence a fiber, because it is the intersection of two saturated
sets: $K$ and the closed ball of radius $\|v\|$ around the origin. ∎

###  2.2. Convex sets and support functions

We will use the concepts of “support function” and “polar” (see e.g. [SSS11,
pages 280–281]), as well as a couple of basic facts.

######  Definition 9.

Let $A\subset V$ be a subset of a finite-dimensional real vector space with
inner product. The _support function_ $h(A,\cdot)\colon
V\to\mathds{R}\cup\infty$ is defined by

$h(A,v)=\sup_{a\in A}\langle v,a\rangle,$

and the _polar_ of $A$ is the subset $A^{\circ}\subset V$ defined by

$A^{\circ}=\\{v\in V\mid h(A,v)\leq 1\\}=\\{v\in V\mid\langle v,a\rangle\leq
1\quad\forall a\in A\\}.$

######  Proposition 10.

Let $A\subset V$ be a closed subset of a finite-dimensional real vector space
with inner product.

  1. (a)

(Bipolar Theorem) If $\operatorname{conv}(A)$ contains the origin, then
$A^{\circ\circ}=\operatorname{conv}(A)$.

  2. (b)

If $\Sigma\subset V$ is a subspace, and $A$ is compact, convex, and contains
the origin, then $\pi_{\Sigma}(A)=(\Sigma\cap A^{\circ})^{\circ}$, where
$\pi_{\Sigma}\colon V\to V$ denotes orthogonal projection onto $\Sigma$.

###### Proof.

  1. (a)

See [Bar02, IV (1.2) page 144].

  2. (b)

It follows directly from Definition 9 that the support function of
$\pi_{\Sigma}(A)$ is the restriction to $\Sigma$ of the support function of
$A$, and, thus, that $\Sigma\cap A^{\circ}=(\pi_{\Sigma}(A))^{\circ}$.

Since $\pi_{\Sigma}(A)$ is a closed convex set containing the origin, the
Bipolar Theorem implies that
$\pi_{\Sigma}(A)=(\pi_{\Sigma}(A))^{\circ\circ}=(\Sigma\cap
A^{\circ})^{\circ}$.

∎

##  3\. Detecting convexity in the base of a submetry

In this section we provide a proof of Theorem A, about how convexity can be
detected metrically in the quotient, and we also investigate convex hulls of
saturated sets. Some Alexandrov Geometry will be used, see [BBI01, Chapter
10], [BGP92], and [Pet07].

###### Proof of Theorem A.

Let $\tilde{f}\colon V\setminus\sigma^{-1}(S)\to(0,\infty)$ be given by
$\tilde{f}=f\circ\sigma$. It follows from the definition of submetry that
$\tilde{f}$ is the distance function to $\sigma^{-1}(S)$.

For $x\in X\setminus S$, let

$|\nabla^{+}f|(x)=\max\left\\{0,\ \lim\sup_{y\to
x}\frac{f(y)-f(x)}{d(y,x)}\right\\}\in[0,1]$

be the “ascending slope” of $f$ at $x$, and analogously for $\tilde{f}$. These
are at most one because $f,\tilde{f}$ are $1$-Lipschitz.

From the definition of submetry, it follows that
$|\nabla^{+}\tilde{f}|(v)=|\nabla^{+}f|(\sigma(v))$ for every $v\in
V\setminus\sigma^{-1}(S)$, see [KL20, Section 2.5].

Since $X,V$ are Alexandrov spaces, the distance function to any point is semi-
concave, and thus the functions $f,\tilde{f}$, being infima of such functions,
are semi-concave as well. In particular, they have well-defined gradients,
which are elements $\nabla_{x}f\in T_{x}X,\ \nabla_{v}\tilde{f}\in T_{v}V$ of
the respective tangent cones, for all $x\in X\setminus S$ and $v\in
V\setminus\sigma^{-1}(S)$. (See [Pet07, Section 1] for the definitions of
semi-concave functions and their gradients.)

Moreover, the norm $\|\nabla_{x}f\|$ of the gradient $\nabla_{x}f\in T_{x}X$
(that is, the distance to the apex of the cone $T_{x}X$) is the maximum of the
differential $d_{x}f$ on the space of directions $\Sigma_{x}X\subset T_{x}X$,
unless $d_{x}f\leq 0$, in which case $\|\nabla_{x}f\|=0$, see [Pet07, Section
1.3]. Therefore $\|\nabla_{x}f\|$ is equal to the ascending slope
$|\nabla^{+}f|(x)$, and analogously for $\tilde{f}$.

Assume $\sigma^{-1}(S)$ is convex. It is well-known that $\tilde{f}$ is
$C^{1}$ with gradient (in the sense of Calculus) of norm one — see, for
example, [BL06, Section 3.3, Exercise 12, on page 57]. Since for $C^{1}$
functions the standard gradient coincides with the gradient in the sense of
[Pet07, Section 1], it follows that $|\nabla^{+}\tilde{f}|$, and hence
$|\nabla^{+}f|$, is constant equal to one.

For the converse, assume $|\nabla^{+}f|(x)=1$ for all $x\in X\setminus S$.
Claim: For every $v\in V\setminus\sigma^{-1}(S)$, with $\tilde{f}(v)=l>0$,
there is a unique closest point $p(v)\in\sigma^{-1}(S)$ with $\|v-p(v)\|=l$.
Moreover, along the geodesic ray $\gamma(t)=p(v)+t(v-p(v))/l$, one has
$\tilde{f}(\gamma(t))=t$ for all $t\geq 0$.

Assuming the Claim, we show that $\sigma^{-1}(S)$ is convex. If not, there
must exist $P,Q\in\sigma^{-1}(S)$ such that $v:=(P+Q)/2\notin\sigma^{-1}(S)$.
We may assume, without loss of generality, that $\angle
p(v)\hat{v}P\geq\pi/2$. By an elementary Calculus argument (compare Lemma 7),

$\lim_{t\to\infty}\big{(}\|\gamma(t)-v\|-\|\gamma(t)-P\|\big{)}\geq 0.$

On the other hand, for $t\geq l$, one has

$\|\gamma(t)-v\|=t-l=\tilde{f}(\gamma(t))-l=d(\gamma(t),\sigma^{-1}(S))-l\leq\|\gamma(t)-P\|-l$

which implies $l\leq 0$, a contradiction. Therefore $\sigma^{-1}(S)$ is
convex.

Finally, we prove the Claim. Let $v\in V\setminus\sigma^{-1}(S)$, and set
$l=\tilde{f}(v)>0$. Choose a straight line $\gamma\colon\mathds{R}\to V$ such
that $\gamma|_{[0,l]}$ is a (unit-speed) minimizing geodesic from
$\sigma^{-1}(S)$ to $v=\gamma(l)$. Note that $\tilde{f}(\gamma(t))=t$ for
$t\in[0,l]$.

By [Pet07, Proposition 2.1.2], there is a gradient curve
$\alpha\colon[0,\infty)\to V$ for $\tilde{f}$, starting at $v=\alpha(0)$. This
means that, for all $t\in[0,\infty)$, the right tangent vector $\alpha^{+}(t)$
coincides with the gradient $\nabla_{\alpha(t)}\tilde{f}$.

Since $|\nabla^{+}f|(\sigma(\alpha(t)))=1$ by assumption, the gradient
$\nabla_{\alpha(t)}\tilde{f}$ has norm one, so that $\alpha$ is parametrized
by arc length.

Moreover, by definition of gradient,
$(d_{\alpha(t)}\tilde{f})(\alpha^{+}(t))=1$ for all $t\geq 0$. This means that
$\tilde{f}\circ\alpha$ has right-derivative identically equal to one, which
implies that

$\tilde{f}(\alpha(t))=l+t\quad\forall t\geq 0.$

But $l+t$ is the length of the concatenated curve
$\gamma|_{[0,l]}*\alpha|_{[0,t]}$, which implies that this curve minimizes
distance between $\sigma^{-1}(S)$ and $\alpha(t)$, so it must be a line
segment, and therefore $\alpha(t)=\gamma(l+t)$ for all $t\geq 0$. In
particular $\gamma|_{[0,l]}$ and the closest point $p(v)=\gamma(0)$ are
uniquely determined by $v$, and $\tilde{f}(\gamma(t))=t$ for $t\in[l,\infty)$.
∎

######  Remark 11.

A key step in the proof of Theorem A above is the fact that if the distance
function $\tilde{f}$ from the closed subset $\sigma^{-1}(S)\subset V$ has
gradient (in the sense of Alexandrov Geometry [Pet07, Section 1]) of norm one
at every $x\in V\setminus\sigma^{-1}(S)$, then $\sigma^{-1}(S)$ is convex. An
alternative proof of this fact, avoiding gradient curves, goes as follows.

There is an explicit formula (see [Pet07, page 10]) for the differential
$d_{x}\tilde{f}$ of $\tilde{f}$ at $v\in T_{x}V=V$, namely

$d_{x}\tilde{f}(v)=\min_{\xi\in\Uparrow_{x}^{\sigma^{-1}(S)}}\langle-\xi,v\rangle$

where $\Uparrow_{x}^{\sigma^{-1}(S)}$ denotes the (compact) subset of the unit
sphere in $T_{x}V=V$ of all initial vectors of minimizing geodesics from $x$
to $\sigma^{-1}(S)$. This formula can be proved using the First Variation
formula for the arc length, along the same lines as the proof of Berger’s
Lemma (see [CE08, Lemma 6.2]).

The condition $\|\nabla_{x}\tilde{f}\|=1$ is equivalent to $d_{x}\tilde{f}$
having maximum value equal $1$ on the unit sphere, and hence, from the formula
above, to $\Uparrow_{x}^{\sigma^{-1}(S)}$ being a singleton222These conditions
are also equivalent to differentiability of the distance function, compare
with [Sak96, Prop. 4.8] and [KL21, Proposition 4.1].. Thus, for every $x\in
V\setminus\sigma^{-1}(S)$, there is a unique minimizing geodesic from $x$ to
$\sigma^{-1}(S)$, that is, $\sigma^{-1}(S)$ is a set of “infinite reach”. But
a closed subset of Euclidean space has infinite reach if and only if it is
convex, see [KP99, Theorems 6.2.12, 6.2.13].

######  Remark 12.

More generally, the distance function to the closed subset
$\sigma^{-1}(S)\subset V$ has gradient of norm one _near_ $\sigma^{-1}(S)$ (as
opposed to on all of $V\setminus\sigma^{-1}(S)$) if and only if
$\sigma^{-1}(S)\subset V$ has “positive reach” — see [KL21, Proposition 1.3
and 4.1].

Thus, one obtains an analogue of Theorem A for sets of positive reach (instead
of convex). Namely, if one replaces “for every $x\in X\setminus S$” with “for
every $x\in O\setminus S$, for some open subset $O\subset X$ containing $S$”,
one obtains a metric condition on $S$ equivalent to $\sigma^{-1}(S)$ having
positive reach.

###  3.1. Convex hulls and polars

Using support functions (Definition 9), we show that taking the convex hull
(or polar) preserves saturation:

######  Proposition 13.

Given a submetry $\sigma\colon V\to X$, such that $\\{0\\}$ is a fiber, and
$A$ any closed saturated subset of $V$, one has:

  1. (a)

The support function $h(A,\cdot)$ is $\sigma$-basic.

  2. (b)

The convex hull $\operatorname{conv}(A)$ is $\sigma$-saturated.

  3. (c)

The polar $A^{\circ}$ is $\sigma$-saturated.

###### Proof.

  1. (a)

Assume first that $A$ is a single fiber $F$. Then $F$ is contained in a sphere
of radius $r$ centered at the origin, and, using Lemma 7, one obtains, for
every $v\in V$:

$h(F,v)=\sup_{a\in F}r\left(\sup_{t>0}\left(t-d\left(v,{t\over
r}a\right)\right)\right)=r\sup_{t>0}\left(t-d\left(v,{t\over
r}F\right)\right).$

By the Homothetic Transformation Lemma (Lemma 5), the set ${t\over r}F$ is a
fiber for every $t>0$, thus $d\left(\cdot,{t\over r}F\right)$ is a basic
function for all $t>0$. Therefore $h(F,\cdot)$ is a basic function.

For $A$ not necessarily a single fiber, one has $h(A,v)=\sup_{F\subset
A}h(F,v)$ for all $v\in V$, where the supremum is taken over all fibers $F$
contained in $A$. Thus $h(A,\cdot)$ is basic.

  2. (b)

Since $A$ is closed, $\operatorname{conv}(A)$ is the intersection of all half-
spaces that contain $A$. In terms of support functions, this reads

$\operatorname{conv}(A)=\\{x\in V\mid\langle v,x\rangle\leq\lambda\quad\forall
v,\lambda\text{ such that }h(A,v)\leq\lambda\\}.$

By part (a), the support function $h(A,\cdot)$ is basic, and therefore this
can be rewritten as

$\operatorname{conv}(A)=\left\\{x\in V\mid\sup_{w\in F_{v}}\langle
w,x\rangle\leq\lambda\quad\forall v,\lambda\text{ such that
}h(A,v)\leq\lambda\right\\},$

where $F_{v}$ denotes the $\sigma$-fiber containing $v$. But $\sup_{w\in
F_{v}}\langle w,x\rangle$ is exactly $h(L_{v},x)$, and, again by part (a),
$h(L_{v},\cdot)$ is basic. Thus $\operatorname{conv}(A)$ is the intersection
of saturated sets, hence saturated.

  3. (c)

This follows immediately from part (a) and the definition of the polar as a
sub-level set of the support function.

∎

######  Remark 14.

Proposition 13 and Theorem A imply that convex hulls can be detected
metrically in the quotient. More precisely, given a submetry $\sigma\colon
V\to X$, such that $\\{0\\}$ is a fiber, and a closed saturated subset
$A\subset V$, the convex hull $\operatorname{conv}(A)$, being closed and
saturated, equals the smallest closed convex saturated subset of $V$
containing $A$. Using Theorem A, $\operatorname{conv}(A)$ is then the inverse
image $\sigma^{-1}(S)$ of the smallest closed subset $S\subset X$ satisfying
the (purely metric) condition in Theorem A and containing $\sigma(A)$.

If, given $S\subset X$ closed, we define $\operatorname{conv}_{0}(S)$ to be
the unique closed subset $C\subset X$ such that
$\operatorname{conv}(\sigma^{-1}(S))=\sigma^{-1}(C)$, then the map
$\operatorname{conv}_{0}$ depends only on the distance function of $X$. In
particular, in the situation of Corollary B,

$\varphi(\operatorname{conv}_{0}(S))=\operatorname{conv}_{0}(\varphi(S)).$

Of special interest to us in this article is the case where $S$ is a single
point. Then $\varphi$ maps (the $\sigma$-image of) every
“fibertope”333generalization of “orbitope”, see e.g. [SSS11]., that is, convex
set of the form $\operatorname{conv}(F)$, where $F$ is a single
$\sigma$-fiber, to another fibertope in $X^{\prime}$.

##  4\. Fat sections

######  Definition 15.

Given a submetry $\sigma\colon V\to X$, such that $\\{0\\}$ is a fiber, we
call a subspace $\Sigma\subset V$ a _fat section_ for $\sigma$ if
$\sigma|_{\Sigma}\colon\Sigma\to X$ is a submetry.

In terms of equidistant decompositions (see Subsection 2.1), Definition 15 can
be rephrased as the following conditions on $\Sigma$: it meets all fibers, the
decomposition $\mathcal{F}$ of $\Sigma$ given by
$\mathcal{F}=\\{F\cap\Sigma\mid F\text{ fiber of }\sigma\\}$ is equidistant,
and the natural bijection $\Sigma/\mathcal{F}\to X$ is an isometry.

In particular, the following are examples of fat sections:

  * •

$\sigma\colon V\to X=V/G$ is the natural quotient map, where $V$ is a polar
$G$-representation, and $\Sigma$ is a section.

  * •

More generally, $\sigma\colon V\to V/G$, where $V$ is a $G$-representation of
nontrivial “copolarity”, and $\Sigma$ is a “generalized minimal section”. See
[GOT04], and [GL14, Section 2.3]).

  * •

$\sigma\colon V\to V/G$, where $V$ is an effective $G$-representation with
non-trivial principal isotropy group $K$, and $\Sigma$ is the fixed-point set
$\Sigma=V^{K}$. See [GL14, page 62] and references therein.

  * •

the fibers of $\sigma\colon V\to X$ form an isoparametric foliation, and
$\Sigma$ is the normal space to a regular leaf. In this case, $\dim(\Sigma)=2$
and $\sigma|_{\Sigma}\colon\Sigma\to X$ is the quotient map of a dihedral
group action.

We turn to the proof of Theorem C, which is a partial generalization of
Theorem 1 (Kostant’s Theorem), and a full generalization of Corollary 2, to
the case of fat sections. We note that Theorem 1 does not fully generalize to
fat sections, see Section 5 for counter-examples.

Since Theorem C concerns orthogonal projection onto a section, but Corollary B
concerns intersection with the fat section, an important step is to link these
two via Proposition 10 (including the Bipolar Theorem). A technical issue
arises from the fact that Proposition 10 applies only to convex sets
_containing the origin_ , which we address using Lemmas 6 and 8, together with
the following:

######  Lemma 16.

Let $\sigma\colon V\to X$ a submetry with $\\{0\\}$ a fiber, and
$\Sigma\subset V$ a fat section. Denote by $V_{0}\subset V$ (respectively
$\Sigma_{0}\subset\Sigma$) the set of all fixed points, that is, the set of
$v\in V$ (respectively $v\in\Sigma$) such that $\\{v\\}$ is a $\sigma$-fiber
(respectively $\sigma|_{\Sigma}$-fiber). Then $V_{0}=\Sigma_{0}$.

###### Proof.

Since $\sigma|_{\Sigma}$ is onto $X$, one has $V_{0}\subset\Sigma_{0}$.

For the reverse inclusion $\Sigma_{0}\subset V_{0}$, use Corollary B. Indeed,
for every $v\in\Sigma_{0}$, $\sigma|_{\Sigma}^{-1}(\sigma(v))=\\{v\\}$ is
convex, hence $\sigma^{-1}(\sigma(v))$ is also convex, but since it is
contained in a sphere, it must be the singleton $\\{v\\}$. In other words,
$v\in V_{0}$. ∎

###### Proof of Theorem C.

  1. (a)

First note that $F$ is compact (being a closed subset of a sphere), so
$K=\operatorname{conv}(F)$ is a compact, convex, $\sigma$-saturated subset of
$V$, by Proposition 13.

Reduction: We reduce the proof to the case where $K$ contains the origin.

Indeed, by Lemma 8, $K$ contains a fixed point $v\in V$, that is, a point $v$
such that $\\{v\\}$ is a $\sigma$-fiber. By Lemma 16, $v\in\Sigma$. By Lemma
6, the translation map $V\to V$, given by $w\mapsto w-v$, sends
$\sigma$-fibers to $\sigma$-fibers.

Thus $F-v$ is a fiber such that $K-v=\operatorname{conv}(F-v)$ contains the
origin, and, assuming
$(K-v)\cap\Sigma=\pi_{\Sigma}(K-v)=\operatorname{conv}((F-v)\cap\Sigma)$, we
obtain

$K\cap\Sigma=v+(K-v)\cap\Sigma=v+\pi_{\Sigma}(K-v)=\pi_{\Sigma}(K)$

and, similarly, $K\cap\Sigma=\operatorname{conv}(F\cap\Sigma)$. This finishes
the proof of the Reduction.

By Corollary B, the map $A\mapsto\Sigma\cap A$ is a bijection between origin-
containing saturated closed convex subsets of $V$ and $\Sigma$. This bijection
preserves the partial order by inclusion.

By Propositions 13(c) and 10(a), taking the polar is an order-reversing
involution of the set of all origin-containing saturated closed convex subsets
of $V$ (respectively $\Sigma$).

Composing these bijections, the map $A\mapsto(\Sigma\cap A^{\circ})^{\circ}$
is another order-preserving bijection between origin-containing saturated
closed convex subsets of $V$ and $\Sigma$. If $A\subset V$ is a closed ball
centered at the origin, $(\Sigma\cap A^{\circ})^{\circ}$ is closed ball (in
$\Sigma$) of the same radius. Thus, the map $A\mapsto(\Sigma\cap
A^{\circ})^{\circ}$ is also an order-preserving bijection between origin-
containing saturated _compact_ convex subsets of $V$ and $\Sigma$. By
Proposition 10(b), this map coincides with $A\mapsto\pi_{\Sigma}(A)$.

Thus there is $K^{\prime}\subset V$ compact, convex, saturated, origin-
containing, such that $\pi_{\Sigma}(K^{\prime})=K\cap\Sigma$. Since
$K\cap\Sigma\subset\pi_{\Sigma}(K)$, we have $K^{\prime}\subset K$.

Choose any $v\in\Sigma\cap F\subset K\cap\Sigma$, and $v^{\prime}\in
K^{\prime}$ such that $\pi_{\Sigma}(v^{\prime})=v$. Since $v^{\prime}\in
K=\operatorname{conv}(F)$, and $F$ is contained in the sphere of radius
$\|v\|$ around the origin, we have $\|v^{\prime}\|\leq\|v\|$. Thus
$v=v^{\prime}$, and $v\in K^{\prime}$. Since $K^{\prime}$ is saturated, we
obtain $F\subset K^{\prime}$, and, since $K^{\prime}$ is convex, we obtain
$K=\operatorname{conv}(F)\subset K^{\prime}$. Thus $K=K^{\prime}$, and
$K\cap\Sigma=\pi_{\Sigma}(K)$.

The other equation $K\cap\Sigma=\operatorname{conv}(F\cap\Sigma)$ follows from
the fact that $A\mapsto A\cap\Sigma$ preserves “fibertopes”, see Remark 14.

Finally, the last statement is clear:
$\pi_{\Sigma}(F)\subset\pi_{\Sigma}(\operatorname{conv}(F))=\operatorname{conv}(F\cap\Sigma)$.

  2. (b)

The inclusion $K\cap\Sigma\subset\pi_{\Sigma}(K)$ is clear. For the reverse
inclusion, let $v\in\pi_{\Sigma}(K)$. Choose $v^{\prime}\in K$ with
$\pi_{\Sigma}(v^{\prime})=v$, and denote by $F_{v^{\prime}}\subset K$ the
$\sigma$-fiber through $v^{\prime}$. Then, by (a),
$v\in\pi_{\Sigma}(F_{v^{\prime}})\subset\operatorname{conv}(\Sigma\cap
F_{v^{\prime}})\subset K\cap\Sigma$.

  3. (c)

The map $A\mapsto A\cap\Sigma$ is a bijection from the set of all
$\sigma$-saturated subsets of $V$ to the set of all
$\sigma|_{\Sigma}$-saturated subsets of $\Sigma$, with inverse
$B\mapsto\sigma^{-1}(\sigma(B))$. We need to show that both these maps
preserve convexity. The first map $A\mapsto A\cap\Sigma$ clearly does. Let
$B\subset\Sigma$ be convex and $\sigma|_{\Sigma}$-saturated. Define
$K=\operatorname{conv}(\sigma^{-1}(\sigma(B)))$, which is a convex
$\sigma$-saturated set by Proposition 13(b). Using part (b),
$K\cap\Sigma=\pi_{\Sigma}(K)=\operatorname{conv}(\pi_{\Sigma}(\sigma^{-1}(\sigma(B))))$.
Using part (a) and convexity of $B$, for each $\sigma$-fiber
$F\subset\sigma^{-1}(\sigma(B))$, we have
$\pi_{\Sigma}(F)\subset\operatorname{conv}(F\cap\Sigma)\subset B$, and thus
$K\cap\Sigma\subset B$. The reverse inclusion being clear, we obtain
$K\cap\Sigma=B$, and therefore, $K=\sigma^{-1}(\sigma(B))$, that is,
$\sigma^{-1}(\sigma(B))$ is convex.

∎

##  5\. Examples

To illustrate that Theorem C applies more generally than Kostant’s Theorem,
one can point to any representation that is not polar but has non-trivial
copolarity in the sense of [GOT04]. Here is one concrete example (see tables
in [GKW21] for many more):

######  Example 17.

Let $2\leq k\leq n-1$ be integers, $G=\mathsf{O}(n)$ be the orthogonal
group444$\mathsf{O}(n)$ can be replaced with $\mathsf{SO}(n)$ and the orbits
would not change. In other words, the orbits are connected., acting diagonally
on $V=(\mathds{R}^{n})^{\oplus k}$. Then a principal isotropy is
$H=\mathsf{O}(n-k)$ (embedded as a diagonal block), whose fixed point set is
$\Sigma=(\mathds{R}^{k})^{\oplus k}$. The normalizer of $H$ in $G$ is
$N(H)=\mathsf{O}(k)\times\mathsf{O}(n-k)$ (embedded as block-diagonal
matrices). Its action has $\mathsf{O}(n-k)$ as ineffective kernel, so it has
the same orbits as the diagonal action of $\mathsf{O}(k)$ on $\Sigma$. The
orbit spaces $V/G$ and $\Sigma/N=\Sigma/\mathsf{O}(k)$ are isometric, and
$\Sigma$ is a fat section for the natural quotient map $\sigma\colon V\to
V/G$.

Next we illustrate that Theorem 1 does not apply for all submetries with a fat
section. That is, the reverse inclusion in Theorem C(a) does not always hold.

The easiest counter-examples can be found considering polar representations
with _disconnected_ orbits, thus also illustrating the necessity of the
hypothesis that orbits are connected in Theorem 1. At the most extreme, one
can take $G\subset\mathsf{O}(V)$ finite and non-trivial. Then $\Sigma=V$ is a
section, so $\pi_{\Sigma}$ is the identity map, and $\pi_{\Sigma}(G\cdot
v)=G\cdot v$ is not convex unless it is a single point. Similar counter-
examples with $G$ infinite are also easily constructed.

A more interesting example, with connected orbits:

######  Example 18.

Let $V,G,\Sigma,\ldots$ as in Example 17. Assume $k>\frac{2n-1}{3}$. Then, for
a generic $v\in\Sigma$, $\pi_{\Sigma}(G\cdot v)$ is strictly contained in the
orbitope $\mathsf{O}(k)\cdot v$. Indeed, since the
$\mathsf{O}(k)$-representation $(\mathds{R}^{k})^{\oplus k}$ is of real type,
the generic orbitope has non-empty interior, that is, dimension $k^{2}$ (see
[SSS11, pages 279-280]). On the other hand, the principal $G$-orbits have
dimension $\dim\mathsf{O}(n)-\dim\mathsf{O}(n-k)=kn-k(k+1)/2$. When
$k>\frac{2n-1}{3}$, this is smaller than $k^{2}$.

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