Hajnal P., Szabo L. es Totik V. (2015) bizonyitottak a kovetkezo erdekes
transzverzalis tetelt.

Ha $A_1,\ldots,A_n$ és $B_1,\ldots,B_n$ ellentetes koruljarasu konvex sokszogek
a sikon, akkor letezik egy egyenes, amely metszi az $A_1B_1,\ldots,A_nB_n$
szakaszok mindegyiket.

Az eloadasban az allitas magasabb dimenzios valtozatait viszgaljuk.
Analog eredmenyt igazolunk $d$-dimenzioban veges ponthalmazok kozti
rendezesfordito
lekepezesekre (megforditja minden olyan $d$-szimplex itanyitasat, melynek
csucsai az
adott halmazban vannak (a konvexitasi feltetel nelkul)).
Kiderul az is, hogy magasabb dimenzioban a ''diszkret'' es a ''folytonos'' eset
nagyon elteroen viselkedik.
Kozos eredmeny A. Holmsennel és E. Roldan-Pensadoval (2016).


Order-type reversing maps and transversals of segments

P. Hajnal, L. I. Szab\'o and V. Totik (2015) proved the following
pretty nice transversal theorem.

If $A_1,\ldots,A_n$ and $B_1,\ldots,B_n$ are convex polygons in the plane
with opposite orientation, then there exists a line that intersects each of the
segments $A_1B_1,\ldots,A_nB_n$.

In this lecture we shall investigate the higher dimensional versions of the
statement.
We prove analog results in $d$-dimension for order-type reversing maps
(the map reverses the orientation of any $d$-simplex with
vertices from the given set (without the convexity assumption)).
Moreover, it turns out that in higher dimensions the ''discrete'' and the
''continuous''
versions of the problem behave rather differently.
This is a joint work with A. Holmsen and E. Rold\'an-Pensado (2016).