32 citations · 122 across the 43 of their papers we have counts for
9 papers · 1 filter
Theorems of Carathéodory, Helly, and Tverberg without dimension
Karim Adiprasito, Imre Bárány, Nabil H. Mustafa +1
We prove a no-dimensional version of Carathédory's theorem: given an -element set , a point $a \in \conv P$, and an integer , , there is a subse…
A Geometric Lower Bound Theorem
Karim Adiprasito, Eran Nevo, José Alejandro Samper
We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their…
Universality theorems for inscribed polytopes and Delaunay triangulations
Karim A. Adiprasito, Arnau Padrol, Louis Theran
We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semialgebraic se…
Linear embeddings of contractible and collapsible complexes
Karim A. Adiprasito, Bruno Benedetti
(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation,…
The universality theorem for neighborly polytopes
Karim A. Adiprasito, Arnau Padrol
In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of an even-dimensional neighborly polytope. This in particular…
A universality theorem for projectively unique polytopes and a conjecture of Shephard
Karim Alexander Adiprasito, Arnau Padrol
We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytop…
Collapsibility of CAT(0) spaces
Karim Adiprasito, Bruno Benedetti
Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a me…
Infinite curvature on typical convex surfaces
Karim Adiprasito
Solving a long-standing open question in convex geometry, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, w…
Characterization of polytopes via tilings with similar pieces
Karim Adiprasito
Generalizing results by Valette, Zamfirescu and Laczkovich, we will prove that a convex body is a polytope if there are sufficiently many tilings which contain a tile similar t…