Extremal inequalities in analysis and convex geometry

Extremal inequalities form a natural meeting point between analysis and convex geometry. In convex geometry, polarity provides a classical relation between a convex body and its dual and underlies many fundamental inequalities involving volume and other notions of size. This project aims to investigate analogous extremal problems for more general dualities on sets, with particular emphasis on geometric inequalities, where volume is replaced by other suitable measures and corresponding functional and information-theoretic inequalities.<br />

<p dir="ltr"><strong data-end="821" data-start="802">Convex geometry</strong> is a branch of mathematics concerned with the study of convex bodies—compact convex sets with non-empty interior—primarily in Euclidean space. Much of the modern theory developed in close connection with the local theory of Banach spaces, a branch of functional analysis, and lies at the interface of geometry and analysis. Despite the apparent simplicity of this definition, the field reveals a rich and intricate structure, explored through fundamental concepts such as volume, surface area and mixed volumes.</p> <p>A cornerstone of the classical theory is polarity, a duality operation that assigns to a convex body K containing the origin another convex set K°, known as its polar. When K is the unit ball of a norm, K° is the unit ball of the dual norm. This construction underpins many elegant results in the field. For instance, the <strong>Blaschke–Santaló inequality</strong> asserts that, among origin-symmetric convex bodies, the product vol(K)vol(K°) is maximised by ellipsoids. This naturally leads to broader questions: what happens when symmetry is removed, when unbounded sets are admitted, or when Lebesgue measure is replaced by a Gaussian or another weighted measure? Such geometric statements often admit functional analogues, and their study draws on methods from analysis such as symmetrisation, variational arguments and optimal transport.</p> <p>This project builds on this rich theory by moving beyond the standard notion of polarity. Its central aim is to investigate more general duality operations on sets and the geometric inequalities associated with them. Several interrelated directions are possible: establishing Santaló-type inequalities; studying dualities that reverse set inclusion and identifying their extremal or self-dual sets; investigating inequalities involving Minkowski and related sums/combinations of convex bodies; and developing corresponding functional inequalities. The precise direction of the project can be adapted to the student’s background and interests.</p>

<p>To apply for this project you will need to make a formal application for research degree study through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University website</a>. You will need to create a login ID with a username and PIN. </p> <p>•    For <strong>Application type</strong> please select <strong>Research Degrees – Research Postgraduate</strong>. <br /> •    The admission year for this project is <strong>2027/28</strong> Academic Year. <br /> •    You will need to select your <strong>Planned Course of Study</strong> from a drop-down menu. For this project, scroll down and select <strong>PHP Pure Mathematics Full Time.</strong> <br /> •    The project start date for this project is <strong>1 October 2027</strong>, please use this as your <strong>Proposed Start Date of Research</strong>. <br /> •    Please state clearly in the<strong> Research Information Section</strong> that the research degree you wish to be considered for is <strong>Extremal inequalities in analysis and convex geometry</strong> as well as <a href="https://eps.leeds.ac.uk/maths/staff/15110/dr-kasia-wyczesany">Dr Kasia Wyczesany</a> as your proposed supervisor.</p> <p><strong>Please state clearly in the Finance section</strong>, <strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong>.</p> <p>More information on how to apply is available on our website <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">here</a>. You will be required to provide a personal statement which outlines your interest in the project you are applying for, why you have chosen it and how your skills map onto the requirements of the project.</p> <p>We will assess applications continuously as we receive them. We welcome and strongly encourage any potential applicants to contact the supervisor(s) for an informal discussion, prior to applying, and recommend submitting your application early.</p> <p><strong>If you are applying with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<p>This is a non funded PhD project. Applicants are expected to be self funded or to secure external funding.</p> <p>You can find out more about our Funding Opportunities on our <a href="https://phd.leeds.ac.uk/">Postgraduate Research Opportunities</a> website.</p> <p><strong>Important: </strong>Please note that all costs associated with your arrival at Leeds (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source.</p>

<p>For questions related to the application process and eligibility, please contact PGR Admissions by email to <a href="mailto:epspgradmissions@leeds.ac.uk">epspgradmissions@leeds.ac.uk</a></p> <p>For further information about this project, please contact Dr Kasia Wyczesany by email to <a href="mailto:K.B.Wyczesany@leeds.ac.uk">K.B.Wyczesany@leeds.ac.uk</a></p>